noxz-sites

A collection of a builder and various scripts creating the noxz.tech sites
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      1 Mathematics
      2 ===========
      3 Under this topic I will not just show tricks, as the notion of tricks in
      4 mathematics is probably the most damning to the spread of mathematical
      5 knowledge. I will instead explain how these, so called, tricks work using
      6 logic and reason. So this topic is not for the savvy hackers or mathematicians
      7 out there.
      8 
      9 {:: class="toc"}
     10 {- class="toc-title"}Contents{--}
     11 + 1 [Divisibility theorems](#math-div-theorems)
     12   + 1.1 [Divisibility by 1](#math-div-theorems-1)
     13   + 1.2 [Divisibility by 2](#math-div-theorems-2)
     14   + 1.3 [Divisibility by 3](#math-div-theorems-3)
     15   + 1.4 [Divisibility by 4](#math-div-theorems-4)
     16   + 1.5 [Divisibility by 5](#math-div-theorems-5)
     17   + 1.6 [Divisibility by 6](#math-div-theorems-6)
     18   + 1.7 [Divisibility by 7](#math-div-theorems-7)
     19   + 1.8 [Divisibility by 8](#math-div-theorems-8)
     20   + 1.9 [Divisibility by 9](#math-div-theorems-9)
     21   + 1.10 [Divisibility by 10](#math-div-theorems-10)
     22   + 1.11 [Divisibility by 11](#math-div-theorems-11)
     23 + 2 [Fractions](#math-fractions)
     24   + 2.1 [The fraction flip when dividing](#math-fractions-flip)
     25 {::}
     26 
     27 {- id="math-div-theorems"}Divisibility theorems{--}
     28 ---------------------------------------------------
     29 When it comes to divisibility there exists some neat theorems to test a certain
     30 number's different divisibilities, or factors. Following are those theorems and
     31 their proof. Some of the proofs are more trivial than others, such as the proof
     32 for divisibility by 1, 2, 5 and 10. From here on out it's assumed that every
     33 number, when working with divisibility, is an integer.
     34 
     35 What is the smallest number that is divisible by 1 through 10? The answer is
     36 {- class="spoiler"}2520{--}. You can try the different divisibility theorems
     37  below on it.
     38 
     39 ### {- id="math-div-theorems-1"}Divisibility by 1{--}
     40 This theorem is quite easy to remember. Every integer is divisible by 1.
     41 
     42 {- class="math block theorem"}
     43 ..{- class="expression"}
     44 ....{-}1{--}
     45 ....{- class="operator"}∣{--}
     46 ....{- class="variable"}a{--}
     47 ..{--}
     48 ..{- class="operator"}⇔{--}
     49 ..{- class="expression"}
     50 ....{- class="variable"}a{--}
     51 ....{- class="operator"}∈{--}
     52 ....{- class="variable"}ℤ{--}
     53 ..{--}
     54 {--}
     55 
     56 ### {- id="math-div-theorems-2"}Divisibility by 2{--}
     57 It's common knowledge that every even number (numbers ending with an even
     58 number) is divisible by 2. This is because even numbers are multiples of 2. In
     59 short, if a number ends with either 0, 2, 4, 6 or 8 it is divisible by 2.
     60 
     61 Say we have a four digit number
     62 {- class="math"}{- class="variable"}abcd{--}{--}, where
     63  {- class="math"}{- class="variable"}a{--}{--} represents the number of
     64  thousands,
     65  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
     66  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
     67  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
     68  can be represented as
     69 {- class="math"}
     70 ..{- class="expression"}
     71 ....{-}1000{--}
     72 ....{- class="variable"}a{--}
     73 ....{- class="operator"}+{--}
     74 ....{-}100{--}
     75 ....{- class="variable"}b{--}
     76 ....{- class="operator"}+{--}
     77 ....{-}10{--}
     78 ....{- class="variable"}c{--}
     79 ....{- class="operator"}+{--}
     80 ....{- class="variable"}d{--}
     81 ..{--}
     82 {--}. If {- class="math"}{- class="variable"}d{--}{--} is divisible by 2 we can
     83  represent it as an even number
     84 {- class="math"}
     85 ..{- class="expression"}
     86 ....{-}2{--}
     87 ....{- class="variable"}n{--}
     88 ..{--}
     89 {--}, like so:
     90 
     91 {- class="math block"}
     92 ..{- class="expression"}
     93 ....{-}1000{--}
     94 ....{- class="variable"}a{--}
     95 ....{- class="operator"}+{--}
     96 ....{-}100{--}
     97 ....{- class="variable"}b{--}
     98 ....{- class="operator"}+{--}
     99 ....{-}10{--}
    100 ....{- class="variable"}c{--}
    101 ....{- class="operator"}+{--}
    102 ....{-}2{--}
    103 ....{- class="variable"}n{--}
    104 ..{--}
    105 {--}
    106 
    107 {- class="math block"}
    108 ..{- class="expression"}
    109 ....{-}2{--}
    110 ....{- class="fenced parenthesis"}
    111 ......{-}({--}
    112 ......{-}500{--}
    113 ......{- class="variable"}a{--}
    114 ......{- class="operator"}+{--}
    115 ......{-}50{--}
    116 ......{- class="variable"}b{--}
    117 ......{- class="operator"}+{--}
    118 ......{-}5{--}
    119 ......{- class="variable"}c{--}
    120 ......{- class="operator"}+{--}
    121 ......{- class="variable"}n{--}
    122 ......{-}){--}
    123 ....{--}
    124 ..{--}
    125 {--}
    126 
    127 We can now see that the number is divisible by 2 if, and only if, the last
    128 digit is divisible by 2. And so the theorem is proven.
    129 
    130 {- class="math block theorem"}
    131 ..{- class="expression"}
    132 ....{-}2{--}
    133 ....{- class="operator"}∣{--}
    134 ....{- class="variable"}abcd{--}
    135 ..{--}
    136 ..{- class="operator"}⇔{--}
    137 ..{- class="expression"}
    138 ....{-}2{--}
    139 ....{- class="operator"}∣{--}
    140 ....{- class="variable"}d{--}
    141 ..{--}
    142 {--}
    143 
    144 ### {- id="math-div-theorems-3"}Divisibility by 3{--}
    145 The theorem goes that if the sum of all digits in a number is divisible by 3,
    146 the whole number is divisible by 3, i.e.
    147  {- class="math"}
    148 ..{- class="expression"}
    149 ....{-}3{--}
    150 ....{- class="operator"}∣{--}
    151 ....{- class="variable"}abcd{--}
    152 ..{--}
    153 ..{- class="operator"}⇔{--}
    154 ..{- class="expression"}
    155 ....{-}3{--}
    156 ....{- class="operator"}∣{--}
    157 ....{- class="fenced parenthesis"}
    158 ......{-}({--}
    159 ......{- class="variable"}a{--}
    160 ......{- class="operator"}+{--}
    161 ......{- class="variable"}b{--}
    162 ......{- class="operator"}+{--}
    163 ......{- class="variable"}c{--}
    164 ......{- class="operator"}+{--}
    165 ......{- class="variable"}d{--}
    166 ......{-}){--}
    167 ....{--}
    168 ..{--}
    169 {--}. Why is this proposition true?
    170 
    171 Let's use the four digit number
    172 {- class="math"}{- class="variable"}abcd{--}{--}, where
    173  {- class="math"}{- class="variable"}a{--}{--} represents the number of
    174  thousands,
    175  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
    176  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
    177  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
    178  can be represented as
    179 {- class="math"}
    180 ..{- class="expression"}
    181 ....{-}1000{--}
    182 ....{- class="variable"}a{--}
    183 ....{- class="operator"}+{--}
    184 ....{-}100{--}
    185 ....{- class="variable"}b{--}
    186 ....{- class="operator"}+{--}
    187 ....{-}10{--}
    188 ....{- class="variable"}c{--}
    189 ....{- class="operator"}+{--}
    190 ....{- class="variable"}d{--}
    191 ..{--}
    192 {--}. Now let's break out one a, b and c from the first 3 terms, and factor out
    193  3 like so:
    194 
    195 {- class="math block"}
    196 ..{- class="expression"}
    197 ....{-}1000{--}
    198 ....{- class="variable"}a{--}
    199 ....{- class="operator"}+{--}
    200 ....{-}100{--}
    201 ....{- class="variable"}b{--}
    202 ....{- class="operator"}+{--}
    203 ....{-}10{--}
    204 ....{- class="variable"}c{--}
    205 ....{- class="operator"}+{--}
    206 ....{- class="variable"}d{--}
    207 ..{--}
    208 {--}
    209 
    210 {- class="math block"}
    211 ..{- class="expression"}
    212 ....{- class="fenced parenthesis"}
    213 ......{-}({--}
    214 ......{-}999{--}
    215 ......{- class="variable"}a{--}
    216 ......{- class="operator"}+{--}
    217 ......{-}99{--}
    218 ......{- class="variable"}b{--}
    219 ......{- class="operator"}+{--}
    220 ......{-}9{--}
    221 ......{- class="variable"}c{--}
    222 ......{-}){--}
    223 ....{--}
    224 ....{- class="operator"}+{--}
    225 ....{- class="fenced parenthesis"}
    226 ......{-}({--}
    227 ......{- class="variable"}a{--}
    228 ......{- class="operator"}+{--}
    229 ......{- class="variable"}b{--}
    230 ......{- class="operator"}+{--}
    231 ......{- class="variable"}c{--}
    232 ......{- class="operator"}+{--}
    233 ......{- class="variable"}d{--}
    234 ......{-}){--}
    235 ....{--}
    236 ..{--}
    237 {--}
    238 
    239 {- class="math block"}
    240 ..{- class="expression"}
    241 ....{-}3{--}
    242 ....{- class="fenced parenthesis"}
    243 ......{-}({--}
    244 ......{-}333{--}
    245 ......{- class="variable"}a{--}
    246 ......{- class="operator"}+{--}
    247 ......{-}33{--}
    248 ......{- class="variable"}b{--}
    249 ......{- class="operator"}+{--}
    250 ......{-}3{--}
    251 ......{- class="variable"}c{--}
    252 ......{-}){--}
    253 ....{--}
    254 ....{- class="operator"}+{--}
    255 ....{- class="fenced parenthesis"}
    256 ......{-}({--}
    257 ......{- class="variable"}a{--}
    258 ......{- class="operator"}+{--}
    259 ......{- class="variable"}b{--}
    260 ......{- class="operator"}+{--}
    261 ......{- class="variable"}c{--}
    262 ......{- class="operator"}+{--}
    263 ......{- class="variable"}d{--}
    264 ......{-}){--}
    265 ....{--}
    266 ..{--}
    267 {--}
    268 
    269 We can now see that the first term is divisible by 3, and the second term is
    270 divisible by 3 if, and only if, the sum
    271  {- class="math"}
    272 ..{- class="expression"}
    273 ....{- class="fenced parenthesis"}
    274 ......{-}({--}
    275 ......{- class="variable"}a{--}
    276 ......{- class="operator"}+{--}
    277 ......{- class="variable"}b{--}
    278 ......{- class="operator"}+{--}
    279 ......{- class="variable"}c{--}
    280 ......{- class="operator"}+{--}
    281 ......{- class="variable"}d{--}
    282 ......{-}){--}
    283 ....{--}
    284 ..{--}
    285 {--} is divisible by 3. And so the theorem is proven.
    286 
    287 {- class="math block theorem"}
    288 ..{- class="expression"}
    289 ....{-}3{--}
    290 ....{- class="operator"}∣{--}
    291 ....{- class="variable"}abcd{--}
    292 ..{--}
    293 ..{- class="operator"}⇔{--}
    294 ..{- class="expression"}
    295 ....{-}3{--}
    296 ....{- class="operator"}∣{--}
    297 ....{- class="fenced parenthesis"}
    298 ......{-}({--}
    299 ......{- class="variable"}a{--}
    300 ......{- class="operator"}+{--}
    301 ......{- class="variable"}b{--}
    302 ......{- class="operator"}+{--}
    303 ......{- class="variable"}c{--}
    304 ......{- class="operator"}+{--}
    305 ......{- class="variable"}d{--}
    306 ......{-}){--}
    307 ....{--}
    308 ..{--}
    309 {--}
    310 
    311 We have shown that the procedure above will hold for all cases. The procedure
    312 is also recursive. If the sum is to hard to test divisibility for, the
    313 procedure can be repeated, until a smaller sum is reveald. This is rarely
    314 necessary as the divisibility of the sum often is easy to determined.
    315 
    316 ### {- id="math-div-theorems-4"}Divisibility by 4{--}
    317 The theorem goes that if the last two digits of a number is divisible by 4, the
    318 whole number is divisible by 4, i.e.
    319  {- class="math"}
    320 ..{- class="expression"}
    321 ....{-}4{--}
    322 ....{- class="operator"}∣{--}
    323 ....{- class="variable"}abcd{--}
    324 ..{--}
    325 ..{- class="operator"}⇔{--}
    326 ..{- class="expression"}
    327 ....{-}4{--}
    328 ....{- class="operator"}∣{--}
    329 ....{- class="variable"}cd{--}
    330 ..{--}
    331 {--}. Why is this proposition true?
    332 
    333 Let's use the four digit number
    334 {- class="math"}{- class="variable"}abcd{--}{--}, where
    335  {- class="math"}{- class="variable"}a{--}{--} represents the number of
    336  thousands,
    337  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
    338  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
    339  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
    340  can be represented as
    341 {- class="math"}
    342 ..{- class="expression"}
    343 ....{-}1000{--}
    344 ....{- class="variable"}a{--}
    345 ....{- class="operator"}+{--}
    346 ....{-}100{--}
    347 ....{- class="variable"}b{--}
    348 ....{- class="operator"}+{--}
    349 ....{-}10{--}
    350 ....{- class="variable"}c{--}
    351 ....{- class="operator"}+{--}
    352 ....{- class="variable"}d{--}
    353 ..{--}
    354 {--}. Now let's factor out 4 from the first two terms, like so:
    355 
    356 {- class="math block"}
    357 ..{- class="expression"}
    358 ....{-}1000{--}
    359 ....{- class="variable"}a{--}
    360 ....{- class="operator"}+{--}
    361 ....{-}100{--}
    362 ....{- class="variable"}b{--}
    363 ....{- class="operator"}+{--}
    364 ....{-}10{--}
    365 ....{- class="variable"}c{--}
    366 ....{- class="operator"}+{--}
    367 ....{- class="variable"}d{--}
    368 ..{--}
    369 {--}
    370 
    371 {- class="math block"}
    372 ..{- class="expression"}
    373 ....{-}4{--}
    374 ....{- class="fenced parenthesis"}
    375 ......{-}({--}
    376 ......{-}250{--}
    377 ......{- class="variable"}a{--}
    378 ......{- class="operator"}+{--}
    379 ......{-}25{--}
    380 ......{- class="variable"}b{--}
    381 ......{-}){--}
    382 ....{--}
    383 ....{- class="operator"}+{--}
    384 ....{- class="fenced parenthesis"}
    385 ......{-}({--}
    386 ......{-}10{--}
    387 ......{- class="variable"}c{--}
    388 ......{- class="operator"}+{--}
    389 ......{- class="variable"}d{--}
    390 ......{-}){--}
    391 ....{--}
    392 ..{--}
    393 {--}
    394 
    395 We can now see that the first term is divisible by 4, so the whole number is
    396 divisible by 4 if, and only if, the second term is divisible by 4. And so the
    397 theorem is proven.
    398 
    399 {- class="math block theorem"}
    400 ..{- class="expression"}
    401 ....{-}4{--}
    402 ....{- class="operator"}∣{--}
    403 ....{- class="variable"}abcd{--}
    404 ..{--}
    405 ..{- class="operator"}⇔{--}
    406 ..{- class="expression"}
    407 ....{-}4{--}
    408 ....{- class="operator"}∣{--}
    409 ....{- class="variable"}cd{--}
    410 ....{--}
    411 ..{--}
    412 {--}
    413 
    414 ### {- id="math-div-theorems-5"}Divisibility by 5{--}
    415 The theorem goes that if the last digits of a number is divisible by 5, the
    416 whole number is divisible by 4, i.e.
    417  {- class="math"}
    418 ..{- class="expression"}
    419 ....{-}5{--}
    420 ....{- class="operator"}∣{--}
    421 ....{- class="variable"}abcd{--}
    422 ..{--}
    423 ..{- class="operator"}⇔{--}
    424 ..{- class="expression"}
    425 ....{-}5{--}
    426 ....{- class="operator"}∣{--}
    427 ....{- class="variable"}d{--}
    428 ..{--}
    429 {--}. Why is this proposition true?
    430 
    431 Let's use the four digit number
    432 {- class="math"}{- class="variable"}abcd{--}{--}, where
    433  {- class="math"}{- class="variable"}a{--}{--} represents the number of
    434  thousands,
    435  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
    436  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
    437  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
    438  can be represented as
    439 {- class="math"}
    440 ..{- class="expression"}
    441 ....{-}1000{--}
    442 ....{- class="variable"}a{--}
    443 ....{- class="operator"}+{--}
    444 ....{-}100{--}
    445 ....{- class="variable"}b{--}
    446 ....{- class="operator"}+{--}
    447 ....{-}10{--}
    448 ....{- class="variable"}c{--}
    449 ....{- class="operator"}+{--}
    450 ....{- class="variable"}d{--}
    451 ..{--}
    452 {--}. Now let's factor out 5 from the first three terms, like so:
    453 
    454 {- class="math block"}
    455 ..{- class="expression"}
    456 ....{-}1000{--}
    457 ....{- class="variable"}a{--}
    458 ....{- class="operator"}+{--}
    459 ....{-}100{--}
    460 ....{- class="variable"}b{--}
    461 ....{- class="operator"}+{--}
    462 ....{-}10{--}
    463 ....{- class="variable"}c{--}
    464 ....{- class="operator"}+{--}
    465 ....{- class="variable"}d{--}
    466 ..{--}
    467 {--}
    468 
    469 {- class="math block"}
    470 ..{- class="expression"}
    471 ....{-}5{--}
    472 ....{- class="fenced parenthesis"}
    473 ......{-}({--}
    474 ......{-}500{--}
    475 ......{- class="variable"}a{--}
    476 ......{- class="operator"}+{--}
    477 ......{-}50{--}
    478 ......{- class="variable"}b{--}
    479 ......{- class="operator"}+{--}
    480 ......{-}5{--}
    481 ......{- class="variable"}c{--}
    482 ......{-}){--}
    483 ....{--}
    484 ....{- class="operator"}+{--}
    485 ....{- class="fenced parenthesis"}
    486 ......{-}({--}
    487 ......{- class="variable"}d{--}
    488 ......{-}){--}
    489 ....{--}
    490 ..{--}
    491 {--}
    492 
    493 We can now see that the first term is divisible by 5, so the whole number is
    494 divisible by 5 if, and only if, the second term is divisible by 5. And so the
    495 theorem is proven. As the only one digit numbers that are divisible by 5 are 0
    496 and 5, another way of putting it is -- if last digit is 0 or 5, the number is
    497 divisible by 5.
    498 
    499 {- class="math block theorem"}
    500 ..{- class="expression"}
    501 ....{-}5{--}
    502 ....{- class="operator"}∣{--}
    503 ....{- class="variable"}abcd{--}
    504 ..{--}
    505 ..{- class="operator"}⇔{--}
    506 ..{- class="expression"}
    507 ....{-}5{--}
    508 ....{- class="operator"}∣{--}
    509 ....{- class="variable"}d{--}
    510 ....{--}
    511 ..{--}
    512 {--}
    513 
    514 ### {- id="math-div-theorems-6"}Divisibility by 6{--}
    515 This theorem is a combination of the theorem for [*divisibility by
    516 2*](#math-div-theorems-2) and [*divisibility by 3*](#math-div-theorems-3).
    517 
    518 {- class="math block theorem"}
    519 ..{- class="expression"}
    520 ....{-}6{--}
    521 ....{- class="operator"}∣{--}
    522 ....{- class="variable"}abcd{--}
    523 ..{--}
    524 ..{- class="operator"}⇔{--}
    525 ..{- class="expression"}
    526 ....{-}3{--}
    527 ....{- class="operator"}∣{--}
    528 ....{- class="fenced parenthesis"}
    529 ......{-}({--}
    530 ......{- class="variable"}a{--}
    531 ......{- class="operator"}+{--}
    532 ......{- class="variable"}b{--}
    533 ......{- class="operator"}+{--}
    534 ......{- class="variable"}c{--}
    535 ......{- class="operator"}+{--}
    536 ......{- class="variable"}d{--}
    537 ......{-}){--}
    538 ....{--}
    539 ..{--}
    540 ..{- class="operator"}∧{--}
    541 ..{- class="expression"}
    542 ....{-}2{--}
    543 ....{- class="operator"}∣{--}
    544 ....{- class="variable"}d{--}
    545 ..{--}
    546 {--}
    547 
    548 **Note**: Be careful when combining theroems like this. Don't be fooled and try
    549 to combine divisibility theorems for 2 and 4 to get the theorem for 8, as
    550 numbers divisible by 4 always are divisible by 2. The number 4 is for instance
    551 both divisible by 2 and 4, but **not** by 8. Make sure the theorems you combine
    552 doesn't share a factor, like 4 and 2 sharing the factor 2. It's of course
    553 possible to use theoerems 4 and 2 together, but it's not certain in all cases
    554 that the two theorems prove divisibility by 8.
    555 
    556 ### {- id="math-div-theorems-7"}Divisibility by 7{--}
    557 Probably one of the most useful theorems is the theorem of *divisibility by 7*,
    558 as it is recursive (just like the theorems of *divisibility by 3 & 9*). The
    559 theorem states that if the difference between the last digit multiplied by 2
    560 and the remaining digits in a number is divisible by 7, the whole number is
    561 divisible by 7. I'll show the procedure with an example below:
    562 
    563 {- class="math block"}
    564 ..{- class="expression"}
    565 ....{-}7{--}
    566 ....{- class="operator"}∣{--}
    567 ....{-}3423{--}
    568 ....{- class="operator"}?{--}
    569 ..{--}
    570 {--}
    571 
    572 {- class="math block"}
    573 ..{- class="expression"}
    574 ....{-}342{--}
    575 ....{- class="operator"}-{--}
    576 ....{-}3{--}
    577 ....{- class="operator"}×{--}
    578 ....{-}2{--}
    579 ....{- class="operator"}={--}
    580 ....{-}336{--}
    581 ..{--}
    582 {--}
    583 
    584 {- class="math block"}
    585 ..{- class="expression"}
    586 ....{-}33{--}
    587 ....{- class="operator"}-{--}
    588 ....{-}6{--}
    589 ....{- class="operator"}×{--}
    590 ....{-}2{--}
    591 ....{- class="operator"}={--}
    592 ....{-}21{--}
    593 ..{--}
    594 {--}
    595 
    596 {- class="math block"}
    597 ..{- class="expression"}
    598 ....{-}7{--}
    599 ....{- class="operator"}∣{--}
    600 ....{-}21{--}
    601 ..{--}
    602 ..{- class="operator"}⇒{--}
    603 ..{- class="expression"}
    604 ....{-}7{--}
    605 ....{- class="operator"}∣{--}
    606 ....{-}3423{--}
    607 ..{--}
    608 {--}
    609 
    610 Neat! So how and why does it work? For simplicity's sake we use a two digit
    611 number
    612 {- class="math"}
    613 ..{- class="expression"}
    614 ....{- class="variable"}ab{--}
    615 ..{--}
    616 {--}, represented as
    617  {- class="math"}
    618 ..{- class="expression"}
    619 ....{-}10{--}
    620 ....{- class="variable"}a{--}
    621 ....{- class="operator"}+{--}
    622 ....{- class="variable"}b{--}
    623 ..{--}
    624 {--}. The theorem says that if (***A***)
    625  {- class="math"}
    626 ..{-}7{--}
    627 ..{- class="operator"}∣{--}
    628 ..{- class="expression"}
    629 ....{- class="variable"}a{--}
    630 ....{- class="operator"}-{--}
    631 ....{-}2{--}
    632 ....{- class="variable"}b{--}
    633 ..{--}
    634 {--} then (***B***)
    635  {- class="math"}
    636 ..{-}7{--}
    637 ..{- class="operator"}∣{--}
    638 ..{- class="expression"}
    639 ....{-}10{--}
    640 ....{- class="variable"}a{--}
    641 ....{- class="operator"}+{--}
    642 ....{- class="variable"}b{--}
    643 ..{--}
    644 {--}. Let's prove it!
    645 
    646 In order to prove the theorem, we must prove both ***A*** and ***B***. So let's
    647 start with ***A***. If we have
    648 {- class="math"}
    649 ..{- class="expression"}
    650 ....{- class="variable"}a{--}
    651 ....{- class="operator"}-{--}
    652 ....{-}2{--}
    653 ....{- class="variable"}b{--}
    654 ..{--}
    655 {--}, and it's divisible by 7, we know that 7 must be a factor of the
    656 expression. We can now create an equation:
    657 
    658 {- class="math block"}
    659 ..{- class="expression"}
    660 ....{- class="variable"}a{--}
    661 ....{- class="operator"}-{--}
    662 ....{-}2{--}
    663 ....{- class="variable"}b{--}
    664 ..{--}
    665 ..{- class="operator"}={--}
    666 ..{- class="expression"}
    667 ....{-}7{--}
    668 ....{- class="variable"}k{--}
    669 ..{--}
    670 {--}
    671 
    672 Multiply the whole equation with 10, and add one extra
    673 {- class="math"}
    674 ..{- class="expression"}
    675 ....{- class="variable"}b{--}
    676 ..{--}
    677 {--}:
    678 
    679 {- class="math block"}
    680 ..{- class="expression"}
    681 ....{- class="hi"}10{--}
    682 ....{- class="variable"}a{--}
    683 ....{- class="operator"}-{--}
    684 ....{- class="hi"}20{--}
    685 ....{- class="variable"}b{--}
    686 ..{--}
    687 ..{- class="operator"}={--}
    688 ..{- class="expression"}
    689 ....{- class="hi"}70{--}
    690 ....{- class="variable"}k{--}
    691 ..{--}
    692 {--}
    693 
    694 {- class="math block"}
    695 ..{- class="expression"}
    696 ....{-}10{--}
    697 ....{- class="variable"}a{--}
    698 ....{- class="operator"}-{--}
    699 ....{-}20{--}
    700 ....{- class="variable"}b{--}
    701 ....{- class="operator hi"}+{--}
    702 ....{- class="variable hi"}b{--}
    703 ..{--}
    704 ..{- class="operator"}={--}
    705 ..{- class="expression"}
    706 ....{-}70{--}
    707 ....{- class="variable"}k{--}
    708 ....{- class="operator hi"}+{--}
    709 ....{- class="variable hi"}b{--}
    710 ..{--}
    711 {--}
    712 
    713 {- class="math block"}
    714 ..{- class="expression"}
    715 ....{-}10{--}
    716 ....{- class="variable"}a{--}
    717 ....{- class="operator"}-{--}
    718 ....{- class="hi"}19{--}
    719 ....{- class="variable hi"}b{--}
    720 ..{--}
    721 ..{- class="operator"}={--}
    722 ..{- class="expression"}
    723 ....{-}70{--}
    724 ....{- class="variable"}k{--}
    725 ....{- class="operator"}+{--}
    726 ....{- class="variable"}b{--}
    727 ..{--}
    728 {--}
    729 
    730 Now add {- class="math"}
    731 ..{- class="expression"}
    732 ....{-}20{--}
    733 ....{- class="variable"}b{--}
    734 ..{--}
    735 {--} to each side of the equation, and try to factor out 7:
    736 
    737 {- class="math block"}
    738 ..{- class="expression"}
    739 ....{-}10{--}
    740 ....{- class="variable"}a{--}
    741 ....{- class="operator"}-{--}
    742 ....{-}19{--}
    743 ....{- class="variable"}b{--}
    744 ....{- class="operator hi"}+{--}
    745 ....{- class="hi"}20{--}
    746 ....{- class="variable hi"}b{--}
    747 ..{--}
    748 ..{- class="operator"}={--}
    749 ..{- class="expression"}
    750 ....{-}70{--}
    751 ....{- class="variable"}k{--}
    752 ....{- class="operator"}+{--}
    753 ....{- class="variable"}b{--}
    754 ....{- class="operator hi"}+{--}
    755 ....{- class="hi"}20{--}
    756 ....{- class="variable hi"}b{--}
    757 ..{--}
    758 {--}
    759 
    760 {- class="math block"}
    761 ..{- class="expression"}
    762 ....{-}10{--}
    763 ....{- class="variable"}a{--}
    764 ....{- class="operator hi"}+{--}
    765 ....{- class="variable hi"}b{--}
    766 ..{--}
    767 ..{- class="operator"}={--}
    768 ..{- class="expression"}
    769 ....{-}70{--}
    770 ....{- class="variable"}k{--}
    771 ....{- class="operator hi"}+{--}
    772 ....{- class="hi"}21{--}
    773 ....{- class="variable hi"}b{--}
    774 ..{--}
    775 {--}
    776 
    777 {- class="math block"}
    778 ..{- class="expression"}
    779 ....{-}10{--}
    780 ....{- class="variable"}a{--}
    781 ....{- class="operator"}+{--}
    782 ....{- class="variable"}b{--}
    783 ..{--}
    784 ..{- class="operator"}={--}
    785 ..{- class="expression hi"}
    786 ....{-}7{--}
    787 ....{- class="fenced parenthesis"}
    788 ......{-}({--}
    789 ......{-}10{--}
    790 ......{- class="variable"}k{--}
    791 ......{- class="operator"}+{--}
    792 ......{-}3{--}
    793 ......{- class="variable"}b{--}
    794 ......{-}){--}
    795 ....{--}
    796 ..{--}
    797 {--}
    798 
    799 We can now see that the right side of the equation is divisible by 7, and our
    800 left side says
    801 {- class="math"}
    802 ..{- class="expression"}
    803 ....{-}10{--}
    804 ....{- class="variable"}a{--}
    805 ....{- class="operator"}+{--}
    806 ....{- class="variable"}b{--}
    807 ..{--}
    808 {--}. Neat, we now know that the two digit number
    809  {- class="math"}
    810 ..{- class="expression"}
    811 ....{- class="variable"}ab{--}
    812 ..{--}
    813 {--} is divisible by 7. Now we must show that ***B*** implies ***A***. That is
    814  if
    815  {- class="math"}
    816 ..{- class="expression"}
    817 ....{- class="variable"}a{--}
    818 ....{- class="operator"}-{--}
    819 ....{-}2{--}
    820 ....{- class="variable"}b{--}
    821 ..{--} is divisible by 7, then
    822  {- class="math"}
    823 ..{- class="expression"}
    824 ....{-}10{--}
    825 ....{- class="variable"}a{--}
    826 ....{- class="operator"}+{--}
    827 ....{- class="variable"}b{--}
    828 ..{--}
    829 {--} is divisible by 7. Let's prove ***B***.
    830 
    831 Just as for ***A***, we know that 7 must be a factor of the expression. We can
    832 now create another equation:
    833 
    834 {- class="math block"}
    835 ..{- class="expression"}
    836 ....{-}10{--}
    837 ....{- class="variable"}a{--}
    838 ....{- class="operator"}+{--}
    839 ....{- class="variable"}b{--}
    840 ..{--}
    841 ..{- class="operator"}={--}
    842 ..{- class="expression"}
    843 ....{-}7{--}
    844 ....{- class="variable"}k{--}
    845 ..{--}
    846 {--}
    847 
    848 Subtract
    849 {- class="math"}
    850 ..{- class="expression"}
    851 ....{-}21{--}
    852 ....{- class="variable"}b{--}
    853 ..{--}
    854 {--} from the whole equation, and factorize:
    855 
    856 {- class="math block"}
    857 ..{- class="expression"}
    858 ....{-}10{--}
    859 ....{- class="variable"}a{--}
    860 ....{- class="operator"}+{--}
    861 ....{- class="variable"}b{--}
    862 ....{- class="operator hi"}-{--}
    863 ....{- class="hi"}21{--}
    864 ....{- class="variable hi"}b{--}
    865 ..{--}
    866 ..{- class="operator"}={--}
    867 ..{- class="expression"}
    868 ....{-}7{--}
    869 ....{- class="variable"}k{--}
    870 ....{- class="operator hi"}-{--}
    871 ....{- class="hi"}21{--}
    872 ....{- class="variable hi"}b{--}
    873 ..{--}
    874 {--}
    875 
    876 {- class="math block"}
    877 ..{- class="expression"}
    878 ....{-}10{--}
    879 ....{- class="variable"}a{--}
    880 ....{- class="operator hi"}-{--}
    881 ....{- class="hi"}20{--}
    882 ....{- class="variable hi"}b{--}
    883 ..{--}
    884 ..{- class="operator"}={--}
    885 ..{- class="expression"}
    886 ....{-}7{--}
    887 ....{- class="variable"}k{--}
    888 ....{- class="operator"}-{--}
    889 ....{-}21{--}
    890 ....{- class="variable"}b{--}
    891 ..{--}
    892 {--}
    893 
    894 {- class="math block"}
    895 ..{- class="expression hi"}
    896 ....{-}10{--}
    897 ....{- class="fenced parenthesis"}
    898 ......{-}({--}
    899 ......{- class="variable"}a{--}
    900 ......{- class="operator"}-{--}
    901 ......{-}2{--}
    902 ......{- class="variable"}b{--}
    903 ......{-}){--}
    904 ....{--}
    905 ..{--}
    906 ..{- class="operator"}={--}
    907 ..{- class="expression hi"}
    908 ....{-}7{--}
    909 ....{- class="fenced parenthesis"}
    910 ......{-}({--}
    911 ......{- class="variable"}k{--}
    912 ......{- class="operator"}-{--}
    913 ......{-}3{--}
    914 ......{- class="variable"}b{--}
    915 ......{-}){--}
    916 ....{--}
    917 ..{--}
    918 {--}
    919 
    920 We can now see that the right side of the equation is divisible by 7, and on
    921 our left side 10 is not divisible by 7 so the expression inside the
    922 parenthesis must be. But isn't that expression
    923 {- class="math"}
    924 ..{- class="expression"}
    925 ....{- class="variable"}a{--}
    926 ....{- class="operator"}-{--}
    927 ....{-}2{--}
    928 ....{- class="variable"}b{--}
    929 ..{--}
    930 {--}. Neat, we now have the proof for the theorem and can conclude that
    931  indeed:
    932 
    933 {- class="math block theorem"}
    934 ..{- class="expression"}
    935 ....{-}7{--}
    936 ....{- class="operator"}∣{--}
    937 ....{- class="expression"}
    938 ......{- class="variable"}ab{--}
    939 ....{--}
    940 ..{--}
    941 ..{- class="operator"}⇔{--}
    942 ..{- class="expression"}
    943 ....{-}7{--}
    944 ....{- class="operator"}∣{--}
    945 ....{- class="expression"}
    946 ......{- class="variable"}a{--}
    947 ......{- class="operator"}-{--}
    948 ......{-}2{--}
    949 ......{- class="variable"}b{--}
    950 ....{--}
    951 ..{--}
    952 {--}
    953 
    954 We have shown that the procedure above will hold for all cases.
    955 
    956 ### {- id="math-div-theorems-8"}Divisibility by 8{--}
    957 The theorem is quite similar to the theorem for *divisibility by 4*. The
    958 theorem goes that if the last three digits of a number is divisible by 8, the
    959 whole number is divisible by 8, i.e.
    960  {- class="math"}
    961 ..{- class="expression"}
    962 ....{-}8{--}
    963 ....{- class="operator"}∣{--}
    964 ....{- class="variable"}abcd{--}
    965 ..{--}
    966 ..{- class="operator"}⇔{--}
    967 ..{- class="expression"}
    968 ....{-}8{--}
    969 ....{- class="operator"}∣{--}
    970 ....{- class="variable"}bcd{--}
    971 ..{--}
    972 {--}. Why is this proposition true?
    973 
    974 Let's use the four digit number
    975 {- class="math"}{- class="variable"}abcd{--}{--}, where
    976  {- class="math"}{- class="variable"}a{--}{--} represents the number of
    977  thousands,
    978  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
    979  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
    980  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
    981  can be represented as
    982 {- class="math"}
    983 ..{- class="expression"}
    984 ....{-}1000{--}
    985 ....{- class="variable"}a{--}
    986 ....{- class="operator"}+{--}
    987 ....{-}100{--}
    988 ....{- class="variable"}b{--}
    989 ....{- class="operator"}+{--}
    990 ....{-}10{--}
    991 ....{- class="variable"}c{--}
    992 ....{- class="operator"}+{--}
    993 ....{- class="variable"}d{--}
    994 ..{--}
    995 {--}. Now let's factor out 8 from the first three terms, like so:
    996 
    997 {- class="math block"}
    998 ..{- class="expression"}
    999 ....{-}1000{--}
   1000 ....{- class="variable"}a{--}
   1001 ....{- class="operator"}+{--}
   1002 ....{-}100{--}
   1003 ....{- class="variable"}b{--}
   1004 ....{- class="operator"}+{--}
   1005 ....{-}10{--}
   1006 ....{- class="variable"}c{--}
   1007 ....{- class="operator"}+{--}
   1008 ....{- class="variable"}d{--}
   1009 ..{--}
   1010 {--}
   1011 
   1012 {- class="math block"}
   1013 ..{- class="expression"}
   1014 ....{-}8{--}
   1015 ....{- class="fenced parenthesis"}
   1016 ......{-}({--}
   1017 ......{-}125{--}
   1018 ......{- class="variable"}a{--}
   1019 ......{-}){--}
   1020 ....{--}
   1021 ....{- class="operator"}+{--}
   1022 ....{- class="fenced parenthesis"}
   1023 ......{-}({--}
   1024 ......{-}100{--}
   1025 ......{- class="variable"}b{--}
   1026 ......{- class="operator"}+{--}
   1027 ......{-}10{--}
   1028 ......{- class="variable"}c{--}
   1029 ......{- class="operator"}+{--}
   1030 ......{- class="variable"}d{--}
   1031 ......{-}){--}
   1032 ....{--}
   1033 ..{--}
   1034 {--}
   1035 
   1036 We can now see that the first term is divisible by 8, so the whole number is
   1037 divisible by 8 if, and only if, the second term is divisible by 8. And so the
   1038 theorem is proven.
   1039 
   1040 {- class="math block theorem"}
   1041 ..{- class="expression"}
   1042 ....{-}8{--}
   1043 ....{- class="operator"}∣{--}
   1044 ....{- class="variable"}abcd{--}
   1045 ..{--}
   1046 ..{- class="operator"}⇔{--}
   1047 ..{- class="expression"}
   1048 ....{-}8{--}
   1049 ....{- class="operator"}∣{--}
   1050 ....{- class="variable"}bcd{--}
   1051 ....{--}
   1052 ..{--}
   1053 {--}
   1054 
   1055 ### {- id="math-div-theorems-9"}Divisibility by 9{--}
   1056 Much like the theorem for *divisibility by 3*, the theorem goes that if the sum
   1057 of all digits in a number is divisible by 9, the whole number is divisible by
   1058 9, i.e.
   1059  {- class="math"}
   1060 ..{- class="expression"}
   1061 ....{-}9{--}
   1062 ....{- class="operator"}∣{--}
   1063 ....{- class="variable"}abcd{--}
   1064 ..{--}
   1065 ..{- class="operator"}⇔{--}
   1066 ..{- class="expression"}
   1067 ....{-}9{--}
   1068 ....{- class="operator"}∣{--}
   1069 ....{- class="fenced parenthesis"}
   1070 ......{-}({--}
   1071 ......{- class="variable"}a{--}
   1072 ......{- class="operator"}+{--}
   1073 ......{- class="variable"}b{--}
   1074 ......{- class="operator"}+{--}
   1075 ......{- class="variable"}c{--}
   1076 ......{- class="operator"}+{--}
   1077 ......{- class="variable"}d{--}
   1078 ......{-}){--}
   1079 ....{--}
   1080 ..{--}
   1081 {--}. Why is this proposition true?
   1082 
   1083 Let's use the four digit number
   1084 {- class="math"}{- class="variable"}abcd{--}{--}, where
   1085  {- class="math"}{- class="variable"}a{--}{--} represents the number of
   1086  thousands,
   1087  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
   1088  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
   1089  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
   1090  can be represented as
   1091 {- class="math"}
   1092 ..{- class="expression"}
   1093 ....{-}1000{--}
   1094 ....{- class="variable"}a{--}
   1095 ....{- class="operator"}+{--}
   1096 ....{-}100{--}
   1097 ....{- class="variable"}b{--}
   1098 ....{- class="operator"}+{--}
   1099 ....{-}10{--}
   1100 ....{- class="variable"}c{--}
   1101 ....{- class="operator"}+{--}
   1102 ....{- class="variable"}d{--}
   1103 ..{--}
   1104 {--}. Now let's break out one a, b and c from the first 9 terms, and factor out
   1105  9 like so:
   1106 
   1107 {- class="math block"}
   1108 ..{- class="expression"}
   1109 ....{-}1000{--}
   1110 ....{- class="variable"}a{--}
   1111 ....{- class="operator"}+{--}
   1112 ....{-}100{--}
   1113 ....{- class="variable"}b{--}
   1114 ....{- class="operator"}+{--}
   1115 ....{-}10{--}
   1116 ....{- class="variable"}c{--}
   1117 ....{- class="operator"}+{--}
   1118 ....{- class="variable"}d{--}
   1119 ..{--}
   1120 {--}
   1121 
   1122 {- class="math block"}
   1123 ..{- class="expression"}
   1124 ....{- class="fenced parenthesis"}
   1125 ......{-}({--}
   1126 ......{-}999{--}
   1127 ......{- class="variable"}a{--}
   1128 ......{- class="operator"}+{--}
   1129 ......{-}99{--}
   1130 ......{- class="variable"}b{--}
   1131 ......{- class="operator"}+{--}
   1132 ......{-}9{--}
   1133 ......{- class="variable"}c{--}
   1134 ......{-}){--}
   1135 ....{--}
   1136 ....{- class="operator"}+{--}
   1137 ....{- class="fenced parenthesis"}
   1138 ......{-}({--}
   1139 ......{- class="variable"}a{--}
   1140 ......{- class="operator"}+{--}
   1141 ......{- class="variable"}b{--}
   1142 ......{- class="operator"}+{--}
   1143 ......{- class="variable"}c{--}
   1144 ......{- class="operator"}+{--}
   1145 ......{- class="variable"}d{--}
   1146 ......{-}){--}
   1147 ....{--}
   1148 ..{--}
   1149 {--}
   1150 
   1151 {- class="math block"}
   1152 ..{- class="expression"}
   1153 ....{-}9{--}
   1154 ....{- class="fenced parenthesis"}
   1155 ......{-}({--}
   1156 ......{-}111{--}
   1157 ......{- class="variable"}a{--}
   1158 ......{- class="operator"}+{--}
   1159 ......{-}11{--}
   1160 ......{- class="variable"}b{--}
   1161 ......{- class="operator"}+{--}
   1162 ......{-}{--}
   1163 ......{- class="variable"}c{--}
   1164 ......{-}){--}
   1165 ....{--}
   1166 ....{- class="operator"}+{--}
   1167 ....{- class="fenced parenthesis"}
   1168 ......{-}({--}
   1169 ......{- class="variable"}a{--}
   1170 ......{- class="operator"}+{--}
   1171 ......{- class="variable"}b{--}
   1172 ......{- class="operator"}+{--}
   1173 ......{- class="variable"}c{--}
   1174 ......{- class="operator"}+{--}
   1175 ......{- class="variable"}d{--}
   1176 ......{-}){--}
   1177 ....{--}
   1178 ..{--}
   1179 {--}
   1180 
   1181 We can now see that the first term is divisible by 9, and the second term is
   1182 divisible by 9 if, and only if, the sum
   1183  {- class="math"}
   1184 ..{- class="expression"}
   1185 ....{- class="fenced parenthesis"}
   1186 ......{-}({--}
   1187 ......{- class="variable"}a{--}
   1188 ......{- class="operator"}+{--}
   1189 ......{- class="variable"}b{--}
   1190 ......{- class="operator"}+{--}
   1191 ......{- class="variable"}c{--}
   1192 ......{- class="operator"}+{--}
   1193 ......{- class="variable"}d{--}
   1194 ......{-}){--}
   1195 ....{--}
   1196 ..{--}
   1197 {--} is divisible by 9. And so the theorem is proven.
   1198 
   1199 {- class="math block theorem"}
   1200 ..{- class="expression"}
   1201 ....{-}9{--}
   1202 ....{- class="operator"}∣{--}
   1203 ....{- class="variable"}abcd{--}
   1204 ..{--}
   1205 ..{- class="operator"}⇔{--}
   1206 ..{- class="expression"}
   1207 ....{-}9{--}
   1208 ....{- class="operator"}∣{--}
   1209 ....{- class="fenced parenthesis"}
   1210 ......{-}({--}
   1211 ......{- class="variable"}a{--}
   1212 ......{- class="operator"}+{--}
   1213 ......{- class="variable"}b{--}
   1214 ......{- class="operator"}+{--}
   1215 ......{- class="variable"}c{--}
   1216 ......{- class="operator"}+{--}
   1217 ......{- class="variable"}d{--}
   1218 ......{-}){--}
   1219 ....{--}
   1220 ..{--}
   1221 {--}
   1222 
   1223 We have shown that the procedure above will hold for all cases. The procedure
   1224 is also recursive. If the sum is to hard to test divisibility for, the
   1225 procedure can be repeated, until a smaller sum is reveald. This is rarely
   1226 necessary as the divisibility of the sum often is easy to determined.
   1227 
   1228 ### {- id="math-div-theorems-10"}Divisibility by 10{--}
   1229 The theorem goes that if the last digits of a number is divisible by 10, the
   1230 whole number is divisible by 10, i.e.
   1231  {- class="math"}
   1232 ..{- class="expression"}
   1233 ....{-}10{--}
   1234 ....{- class="operator"}∣{--}
   1235 ....{- class="variable"}abcd{--}
   1236 ..{--}
   1237 ..{- class="operator"}⇔{--}
   1238 ..{- class="expression"}
   1239 ....{-}10{--}
   1240 ....{- class="operator"}∣{--}
   1241 ....{- class="variable"}d{--}
   1242 ..{--}
   1243 {--}. Why is this proposition true?
   1244 
   1245 Let's use the four digit number
   1246 {- class="math"}{- class="variable"}abcd{--}{--}, where
   1247  {- class="math"}{- class="variable"}a{--}{--} represents the number of
   1248  thousands,
   1249  {- class="math"}{- class="variable"}b{--}{--} the number of hundreds,
   1250  {- class="math"}{- class="variable"}c{--}{--} the number of tens and
   1251  {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number
   1252  can be represented as
   1253 {- class="math"}
   1254 ..{- class="expression"}
   1255 ....{-}1000{--}
   1256 ....{- class="variable"}a{--}
   1257 ....{- class="operator"}+{--}
   1258 ....{-}100{--}
   1259 ....{- class="variable"}b{--}
   1260 ....{- class="operator"}+{--}
   1261 ....{-}10{--}
   1262 ....{- class="variable"}c{--}
   1263 ....{- class="operator"}+{--}
   1264 ....{- class="variable"}d{--}
   1265 ..{--}
   1266 {--}. Now let's factor out 10 from the first three terms, like so:
   1267 
   1268 {- class="math block"}
   1269 ..{- class="expression"}
   1270 ....{-}1000{--}
   1271 ....{- class="variable"}a{--}
   1272 ....{- class="operator"}+{--}
   1273 ....{-}100{--}
   1274 ....{- class="variable"}b{--}
   1275 ....{- class="operator"}+{--}
   1276 ....{-}10{--}
   1277 ....{- class="variable"}c{--}
   1278 ....{- class="operator"}+{--}
   1279 ....{- class="variable"}d{--}
   1280 ..{--}
   1281 {--}
   1282 
   1283 {- class="math block"}
   1284 ..{- class="expression"}
   1285 ....{-}10{--}
   1286 ....{- class="fenced parenthesis"}
   1287 ......{-}({--}
   1288 ......{-}100{--}
   1289 ......{- class="variable"}a{--}
   1290 ......{- class="operator"}+{--}
   1291 ......{-}10{--}
   1292 ......{- class="variable"}b{--}
   1293 ......{- class="operator"}+{--}
   1294 ......{- class="variable"}c{--}
   1295 ......{-}){--}
   1296 ....{--}
   1297 ....{- class="operator"}+{--}
   1298 ....{- class="fenced parenthesis"}
   1299 ......{-}({--}
   1300 ......{- class="variable"}d{--}
   1301 ......{-}){--}
   1302 ....{--}
   1303 ..{--}
   1304 {--}
   1305 
   1306 We can now see that the first term is divisible by 10, so the whole number is
   1307 divisible by 10 if, and only if, the second term is divisible by 10. And so the
   1308 theorem is proven. As the only one digit number that is divisible by 10 is 0,
   1309 another way of putting it is -- if last digit is 0, the number is
   1310 divisible by 10.
   1311 
   1312 {- class="math block theorem"}
   1313 ..{- class="expression"}
   1314 ....{-}10{--}
   1315 ....{- class="operator"}∣{--}
   1316 ....{- class="variable"}abcd{--}
   1317 ..{--}
   1318 ..{- class="operator"}⇔{--}
   1319 ..{- class="expression"}
   1320 ....{-}10{--}
   1321 ....{- class="operator"}∣{--}
   1322 ....{- class="variable"}d{--}
   1323 ....{--}
   1324 ..{--}
   1325 {--}
   1326 
   1327 ### {- id="math-div-theorems-11"}Divisibility by 11{--}
   1328 The theorem goes that a number is divisible by 11 if, and only if, the
   1329 alternate sum of its digits is divisible by 11, like so:
   1330 
   1331 {- class="math block"}
   1332 ..{- class="expression"}
   1333 ....{-}11{--}
   1334 ....{- class="operator"}∣{--}
   1335 ....{-}190905{--}
   1336 ....{- class="operator"}?{--}
   1337 ..{--}
   1338 {--}
   1339 
   1340 {- class="math block"}
   1341 ..{- class="expression"}
   1342 ....{-}1{--}
   1343 ....{- class="operator"}-{--}
   1344 ....{-}9{--}
   1345 ....{- class="operator"}+{--}
   1346 ....{-}0{--}
   1347 ....{- class="operator"}-{--}
   1348 ....{-}9{--}
   1349 ....{- class="operator"}+{--}
   1350 ....{-}0{--}
   1351 ....{- class="operator"}-{--}
   1352 ....{-}5{--}
   1353 ....{- class="operator"}={--}
   1354 ....{-}-22{--}
   1355 ..{--}
   1356 {--}
   1357 
   1358 {- class="math block"}
   1359 ..{- class="expression"}
   1360 ....{-}11{--}
   1361 ....{- class="operator"}∣{--}
   1362 ....{-}-22{--}
   1363 ..{--}
   1364 ..{- class="operator"}⇒{--}
   1365 ..{- class="expression"}
   1366 ....{-}11{--}
   1367 ....{- class="operator"}∣{--}
   1368 ....{-}190905{--}
   1369 ..{--}
   1370 {--}
   1371 
   1372 Neat! So how and why does it work? For simplicity's sake we use a four digit
   1373 number
   1374 {- class="math"}
   1375 ..{- class="expression"}
   1376 ....{- class="variable"}abcd{--}
   1377 ..{--}
   1378 {--}, represented as
   1379  {- class="math"}
   1380 ..{- class="expression"}
   1381 ....{-}1000{--}
   1382 ....{- class="variable"}a{--}
   1383 ....{- class="operator"}+{--}
   1384 ....{-}100{--}
   1385 ....{- class="variable"}b{--}
   1386 ....{- class="operator"}+{--}
   1387 ....{-}10{--}
   1388 ....{- class="variable"}c{--}
   1389 ....{- class="operator"}+{--}
   1390 ....{- class="variable"}d{--}
   1391 ..{--}
   1392 {--}. This expression can also be represented in another way by manipulating
   1393 the terms. We give and take in an alternating fashion, like so:
   1394 
   1395 {- class="math block"}
   1396 ..{- class="expression"}
   1397 ....{-}1000{--}
   1398 ....{- class="variable"}a{--}
   1399 ....{- class="operator"}+{--}
   1400 ....{-}100{--}
   1401 ....{- class="variable"}b{--}
   1402 ....{- class="operator"}+{--}
   1403 ....{-}10{--}
   1404 ....{- class="variable"}c{--}
   1405 ....{- class="operator"}+{--}
   1406 ....{- class="variable"}d{--}
   1407 ..{--}
   1408 {--}
   1409 
   1410 {- class="math block"}
   1411 ..{- class="expression"}
   1412 ....{- class="variable hi"}a{--}
   1413 ....{- class="fenced parenthesis hi"}
   1414 ......{-}({--}
   1415 ......{-}1000{--}
   1416 ......{-}){--}
   1417 ....{--}
   1418 ....{- class="operator"}+{--}
   1419 ....{- class="variable hi"}b{--}
   1420 ....{- class="fenced parenthesis hi"}
   1421 ......{-}({--}
   1422 ......{-}100{--}
   1423 ......{-}){--}
   1424 ....{--}
   1425 ....{- class="operator"}+{--}
   1426 ....{- class="variable hi"}c{--}
   1427 ....{- class="fenced parenthesis hi"}
   1428 ......{-}({--}
   1429 ......{-}10{--}
   1430 ......{-}){--}
   1431 ....{--}
   1432 ....{- class="operator"}+{--}
   1433 ....{- class="variable"}d{--}
   1434 ..{--}
   1435 {--}
   1436 
   1437 {- class="math block"}
   1438 ..{- class="expression"}
   1439 ....{- class="variable"}a{--}
   1440 ....{- class="fenced parenthesis"}
   1441 ......{-}({--}
   1442 ......{- class="hi"}1001{--}
   1443 ......{- class="operator hi"}-{--}
   1444 ......{- class="hi"}1{--}
   1445 ......{-}){--}
   1446 ....{--}
   1447 ....{- class="operator"}+{--}
   1448 ....{- class="variable"}b{--}
   1449 ....{- class="fenced parenthesis"}
   1450 ......{-}({--}
   1451 ......{- class="hi"}99{--}
   1452 ......{- class="operator hi"}+{--}
   1453 ......{- class="hi"}1{--}
   1454 ......{-}){--}
   1455 ....{--}
   1456 ....{- class="operator"}+{--}
   1457 ....{- class="variable"}c{--}
   1458 ....{- class="fenced parenthesis"}
   1459 ......{-}({--}
   1460 ......{- class="hi"}11{--}
   1461 ......{- class="operator hi"}-{--}
   1462 ......{- class="hi"}1{--}
   1463 ......{-}){--}
   1464 ....{--}
   1465 ....{- class="operator"}+{--}
   1466 ....{- class="variable"}d{--}
   1467 ..{--}
   1468 {--}
   1469 
   1470 {- class="math block"}
   1471 ..{- class="expression"}
   1472 ....{- class="hi"}1001{--}
   1473 ....{- class="variable hi"}a{--}
   1474 ....{- class="operator hi"}-{--}
   1475 ....{- class="variable hi"}a{--}
   1476 ....{- class="operator"}+{--}
   1477 ....{- class="hi"}99{--}
   1478 ....{- class="variable hi"}b{--}
   1479 ....{- class="operator hi"}+{--}
   1480 ....{- class="variable hi"}b{--}
   1481 ....{- class="operator"}+{--}
   1482 ....{- class="hi"}11{--}
   1483 ....{- class="variable hi"}c{--}
   1484 ....{- class="operator hi"}-{--}
   1485 ....{- class="variable hi"}c{--}
   1486 ....{- class="operator"}+{--}
   1487 ....{- class="variable"}d{--}
   1488 ..{--}
   1489 {--}
   1490 
   1491 {- class="math block"}
   1492 ..{- class="expression"}
   1493 ....{-}1001{--}
   1494 ....{- class="variable"}a{--}
   1495 ....{- class="operator hi"}+{--}
   1496 ....{- class="hi"}99{--}
   1497 ....{- class="variable hi"}b{--}
   1498 ....{- class="operator hi"}+{--}
   1499 ....{- class="hi"}11{--}
   1500 ....{- class="variable hi"}c{--}
   1501 ....{- class="operator hi"}-{--}
   1502 ....{- class="variable hi"}a{--}
   1503 ....{- class="operator hi"}+{--}
   1504 ....{- class="variable hi"}b{--}
   1505 ....{- class="operator hi"}-{--}
   1506 ....{- class="variable hi"}c{--}
   1507 ....{- class="operator"}+{--}
   1508 ....{- class="variable"}d{--}
   1509 ..{--}
   1510 {--}
   1511 
   1512 Now we factorize the expression, like so:
   1513 
   1514 {- class="math block"}
   1515 ..{- class="expression"}
   1516 ....{- class="hi"}11{--}
   1517 ....{- class="fenced parenthesis hi"}
   1518 ......{-}({--}
   1519 ......{-}91{--}
   1520 ......{- class="variable"}a{--}
   1521 ......{- class="operator"}+{--}
   1522 ......{-}9{--}
   1523 ......{- class="variable"}b{--}
   1524 ......{- class="operator"}+{--}
   1525 ......{-}1{--}
   1526 ......{- class="variable"}c{--}
   1527 ......{-}){--}
   1528 ....{--}
   1529 ....{- class="operator"}-{--}
   1530 ....{- class="variable"}a{--}
   1531 ....{- class="operator"}+{--}
   1532 ....{- class="variable"}b{--}
   1533 ....{- class="operator"}-{--}
   1534 ....{- class="variable"}c{--}
   1535 ....{- class="operator"}+{--}
   1536 ....{- class="variable"}d{--}
   1537 ..{--}
   1538 {--}
   1539 
   1540 We can see that the first term in the expression is divisible by 11. This means
   1541 that if, and only if, the sum of the other terms is divisible by 11 the whole
   1542 expression is divisible by 11, and so the theorem is proven.
   1543 
   1544 {- class="math block theorem"}
   1545 ..{- class="expression"}
   1546 ....{-}11{--}
   1547 ....{- class="operator"}∣{--}
   1548 ....{- class="variable"}abcd{--}
   1549 ..{--}
   1550 ..{- class="operator"}⇔{--}
   1551 ..{- class="expression"}
   1552 ....{-}11{--}
   1553 ....{- class="operator"}∣{--}
   1554 ....{- class="operator"}-{--}
   1555 ....{- class="variable"}a{--}
   1556 ....{- class="operator"}+{--}
   1557 ....{- class="variable"}b{--}
   1558 ....{- class="operator"}-{--}
   1559 ....{- class="variable"}c{--}
   1560 ....{- class="operator"}+{--}
   1561 ....{- class="variable"}d{--}
   1562 ..{--}
   1563 ..{- class="break"}{--}
   1564 ..{- class="expression"}
   1565 ....{-}11{--}
   1566 ....{- class="operator"}∣{--}
   1567 ....{- class="variable"}abcd{--}
   1568 ..{--}
   1569 ..{- class="operator"}⇔{--}
   1570 ..{- class="expression"}
   1571 ....{-}11{--}
   1572 ....{- class="operator"}∣{--}
   1573 ....{- class="variable"}a{--}
   1574 ....{- class="operator"}-{--}
   1575 ....{- class="variable"}b{--}
   1576 ....{- class="operator"}+{--}
   1577 ....{- class="variable"}c{--}
   1578 ....{- class="operator"}-{--}
   1579 ....{- class="variable"}d{--}
   1580 ..{--}
   1581 {--}
   1582 
   1583 We have shown that the procedure above will hold for all cases, as the number
   1584 can be extended with infinite digits and still follow the same pattern.
   1585 
   1586 {- id="math-fractions"}Fractions{--}
   1587 ------------------------------------
   1588 
   1589 ### {- id="math-fractions-flip"}The fraction flip when dividing{--}
   1590 Many of you have probably been taught the trick of flipping the right fraction
   1591 in a division to instead use simpler multiplication. This is how it works:
   1592 
   1593 We start of with the division of
   1594 {- class="math"}
   1595 ..{- class="fraction"}
   1596 ....{-}3{--}
   1597 ....{-}2{--}
   1598 ..{--}
   1599 {--}
   1600  by
   1601 {- class="math"}
   1602 ..{- class="fraction"}
   1603 ....{-}4{--}
   1604 ....{-}5{--}
   1605 ..{--}
   1606 {--}. From there we can reconstruct the two fractions as the dividend over
   1607  the divisor with a horizontal line, for simplicity's sake, like so:
   1608 
   1609 {- class="math block"}
   1610 ..{- class="fraction"}
   1611 ....{-}3{--}
   1612 ....{-}2{--}
   1613 ..{--}
   1614 ..{- class="operator"}÷{--}
   1615 ..{- class="fraction"}
   1616 ....{-}4{--}
   1617 ....{-}5{--}
   1618 ..{--}
   1619 ..{- class="operator"}={--}
   1620 ..{- class="fraction"}
   1621 ....{-}
   1622 ......{- class="fraction"}
   1623 ........{-}3{--}
   1624 ........{-}2{--}
   1625 ......{--}
   1626 ....{--}
   1627 ....{-}
   1628 ......{- class="fraction"}
   1629 ........{-}4{--}
   1630 ........{-}5{--}
   1631 ......{--}
   1632 ....{--}
   1633 ..{--}
   1634 {--}
   1635 
   1636 After that the "trick" can begin. First we multiply both the dividend and the
   1637 divisor with the inverse of the divisor. As long as we treat the dividend and
   1638 the divisor the same way, this is fine. Be aware of PEMDAS though! If any of
   1639 the dividend or the divisor would have been an addition or subtraction you
   1640 would have to multiply both terms by *x*, either by *x(a + b)* or *xa + xb*.
   1641 
   1642 {- class="math block"}
   1643 ..{- class="fraction"}
   1644 ....{-}3{--}
   1645 ....{-}2{--}
   1646 ..{--}
   1647 ..{- class="operator"}÷{--}
   1648 ..{- class="fraction"}
   1649 ....{-}4{--}
   1650 ....{-}5{--}
   1651 ..{--}
   1652 ..{- class="operator"}={--}
   1653 ..{- class="fraction"}
   1654 ....{-}
   1655 ......{- class="fraction"}
   1656 ........{-}3{--}
   1657 ........{-}2{--}
   1658 ......{--}
   1659 ....{--}
   1660 ....{-}
   1661 ......{- class="fraction"}
   1662 ........{-}4{--}
   1663 ........{-}5{--}
   1664 ......{--}
   1665 ....{--}
   1666 ..{--}
   1667 ..{- class="operator"}={--}
   1668 ..{- class="fraction"}
   1669 ....{-}
   1670 ......{- class="fraction"}
   1671 ........{-}3{--}
   1672 ........{-}2{--}
   1673 ......{--}
   1674 ......{- class="operator hi"}×{--}
   1675 ......{- class="fraction hi"}
   1676 ........{-}5{--}
   1677 ........{-}4{--}
   1678 ......{--}
   1679 ....{--}
   1680 ....{-}
   1681 ......{- class="fraction"}
   1682 ........{-}4{--}
   1683 ........{-}5{--}
   1684 ......{--}
   1685 ......{- class="operator hi"}×{--}
   1686 ......{- class="fraction hi"}
   1687 ........{-}5{--}
   1688 ........{-}4{--}
   1689 ......{--}
   1690 ....{--}
   1691 ..{--}
   1692 ..{- class="operator"}={--}
   1693 ..{- class="fraction"}
   1694 ....{-}
   1695 ......{- class="fraction"}
   1696 ........{-}3{--}
   1697 ........{-}2{--}
   1698 ......{--}
   1699 ......{- class="operator"}×{--}
   1700 ......{- class="fraction"}
   1701 ........{-}5{--}
   1702 ........{-}4{--}
   1703 ......{--}
   1704 ....{--}
   1705 ....{-}
   1706 ......{- class="fraction hi"}
   1707 ........{-}4{- class="operator"}×{--}5{--}
   1708 ........{-}5{- class="operator"}×{--}4{--}
   1709 ......{--}
   1710 ....{--}
   1711 ..{--}
   1712 ..{- class="operator"}={--}
   1713 ..{- class="fraction"}
   1714 ....{-}
   1715 ......{- class="fraction"}
   1716 ........{-}3{--}
   1717 ........{-}2{--}
   1718 ......{--}
   1719 ......{- class="operator"}×{--}
   1720 ......{- class="fraction"}
   1721 ........{-}5{--}
   1722 ........{-}4{--}
   1723 ......{--}
   1724 ....{--}
   1725 ....{-}
   1726 ......{- class="fraction hi"}
   1727 ........{-}20{--}
   1728 ........{-}20{--}
   1729 ......{--}
   1730 ....{--}
   1731 ..{--}
   1732 ..{- class="operator"}={--}
   1733 ..{- class="fraction"}
   1734 ....{-}
   1735 ......{- class="fraction"}
   1736 ........{-}3{--}
   1737 ........{-}2{--}
   1738 ......{--}
   1739 ......{- class="operator"}×{--}
   1740 ......{- class="fraction"}
   1741 ........{-}5{--}
   1742 ........{-}4{--}
   1743 ......{--}
   1744 ....{--}
   1745 ....{-}
   1746 ......{- class="fraction hi"}
   1747 ........{-}1{--}
   1748 ........{- class="hidden"} {--}
   1749 ......{--}
   1750 ....{--}
   1751 ..{--}
   1752 ..{- class="operator"}={--}
   1753 ..{- class="fraction"}
   1754 ....{-}3{--}
   1755 ....{-}2{--}
   1756 ..{--}
   1757 ..{- class="operator"}×{--}
   1758 ..{- class="fraction"}
   1759 ....{-}5{--}
   1760 ....{-}4{--}
   1761 ..{--}
   1762 {--}
   1763 
   1764 As you can see the right fraction has now "flipped", and not by magic, but with
   1765 logic and reason. As division by 1 is equal to the dividend, we can then solve
   1766 the expression, like so:
   1767 
   1768 {- class="math block"}
   1769 ..{- class="fraction"}
   1770 ....{-}3{--}
   1771 ....{-}2{--}
   1772 ..{--}
   1773 ..{- class="operator"}×{--}
   1774 ..{- class="fraction"}
   1775 ....{-}5{--}
   1776 ....{-}4{--}
   1777 ..{--}
   1778 ..{- class="operator"}={--}
   1779 ..{- class="fraction"}
   1780 ....{-}3{- class="operator"}×{--}5{--}
   1781 ....{-}2{- class="operator"}×{--}4{--}
   1782 ..{--}
   1783 ..{- class="operator"}={--}
   1784 ..{- class="fraction"}
   1785 ....{-}15{--}
   1786 ....{-}8{--}
   1787 ..{--}
   1788 {--}
   1789 
   1790 As the final cherry on top, we can prove the procedure by dividing 1 by 2 as we
   1791 know this should result in one half.
   1792 
   1793 {- class="math block"}
   1794 ..{-}1{--}
   1795 ..{- class="operator"}÷{--}
   1796 ..{-}2{--}
   1797 ..{- class="operator"}={--}
   1798 ..{- class="fraction"}
   1799 ....{-}1{--}
   1800 ....{-}1{--}
   1801 ..{--}
   1802 ..{- class="operator"}÷{--}
   1803 ..{- class="fraction"}
   1804 ....{-}2{--}
   1805 ....{-}1{--}
   1806 ..{--}
   1807 ..{- class="operator"}={--}
   1808 ..{- class="fraction"}
   1809 ....{-}
   1810 ......{- class="fraction"}
   1811 ........{-}1{--}
   1812 ........{-}1{--}
   1813 ......{--}
   1814 ....{--}
   1815 ....{-}
   1816 ......{- class="fraction"}
   1817 ........{-}2{--}
   1818 ........{-}1{--}
   1819 ......{--}
   1820 ....{--}
   1821 ..{--}
   1822 {--}
   1823 
   1824 {- class="math block"}
   1825 ..{- class="fraction"}
   1826 ....{-}1{--}
   1827 ....{-}1{--}
   1828 ..{--}
   1829 ..{- class="operator"}÷{--}
   1830 ..{- class="fraction"}
   1831 ....{-}2{--}
   1832 ....{-}1{--}
   1833 ..{--}
   1834 ..{- class="operator"}={--}
   1835 ..{- class="fraction"}
   1836 ....{-}
   1837 ......{- class="fraction"}
   1838 ........{-}1{--}
   1839 ........{-}1{--}
   1840 ......{--}
   1841 ....{--}
   1842 ....{-}
   1843 ......{- class="fraction"}
   1844 ........{-}2{--}
   1845 ........{-}1{--}
   1846 ......{--}
   1847 ....{--}
   1848 ..{--}
   1849 ..{- class="operator"}={--}
   1850 ..{- class="fraction"}
   1851 ....{-}
   1852 ......{- class="fraction"}
   1853 ........{-}1{--}
   1854 ........{-}1{--}
   1855 ......{--}
   1856 ......{- class="operator hi"}×{--}
   1857 ......{- class="fraction hi"}
   1858 ........{-}1{--}
   1859 ........{-}2{--}
   1860 ......{--}
   1861 ....{--}
   1862 ....{-}
   1863 ......{- class="fraction"}
   1864 ........{-}2{--}
   1865 ........{-}1{--}
   1866 ......{--}
   1867 ......{- class="operator hi"}×{--}
   1868 ......{- class="fraction hi"}
   1869 ........{-}1{--}
   1870 ........{-}2{--}
   1871 ......{--}
   1872 ....{--}
   1873 ..{--}
   1874 ..{- class="operator"}={--}
   1875 ..{- class="fraction"}
   1876 ....{-}
   1877 ......{- class="fraction"}
   1878 ........{-}1{--}
   1879 ........{-}1{--}
   1880 ......{--}
   1881 ......{- class="operator"}×{--}
   1882 ......{- class="fraction"}
   1883 ........{-}1{--}
   1884 ........{-}2{--}
   1885 ......{--}
   1886 ....{--}
   1887 ....{-}
   1888 ......{- class="fraction hi"}
   1889 ........{-}2{- class="operator"}×{--}1{--}
   1890 ........{-}1{- class="operator"}×{--}2{--}
   1891 ......{--}
   1892 ....{--}
   1893 ..{--}
   1894 ..{- class="operator"}={--}
   1895 ..{- class="fraction"}
   1896 ....{-}
   1897 ......{- class="fraction"}
   1898 ........{-}1{--}
   1899 ........{-}1{--}
   1900 ......{--}
   1901 ......{- class="operator"}×{--}
   1902 ......{- class="fraction"}
   1903 ........{-}1{--}
   1904 ........{-}2{--}
   1905 ......{--}
   1906 ....{--}
   1907 ....{-}
   1908 ......{- class="fraction hi"}
   1909 ........{-}2{--}
   1910 ........{-}2{--}
   1911 ......{--}
   1912 ....{--}
   1913 ..{--}
   1914 ..{- class="operator"}={--}
   1915 ..{- class="fraction"}
   1916 ....{-}
   1917 ......{- class="fraction"}
   1918 ........{-}1{--}
   1919 ........{-}1{--}
   1920 ......{--}
   1921 ......{- class="operator"}×{--}
   1922 ......{- class="fraction"}
   1923 ........{-}1{--}
   1924 ........{-}2{--}
   1925 ......{--}
   1926 ....{--}
   1927 ....{-}
   1928 ......{- class="fraction hi"}
   1929 ........{-}1{--}
   1930 ........{- class="hidden"} {--}
   1931 ......{--}
   1932 ....{--}
   1933 ..{--}
   1934 ..{- class="operator"}={--}
   1935 ..{- class="fraction"}
   1936 ....{-}1{--}
   1937 ....{-}1{--}
   1938 ..{--}
   1939 ..{- class="operator"}×{--}
   1940 ..{- class="fraction"}
   1941 ....{-}1{--}
   1942 ....{-}2{--}
   1943 ..{--}
   1944 {--}
   1945 
   1946 {- class="math block"}
   1947 ..{- class="fraction"}
   1948 ....{-}1{--}
   1949 ....{-}1{--}
   1950 ..{--}
   1951 ..{- class="operator"}×{--}
   1952 ..{- class="fraction"}
   1953 ....{-}1{--}
   1954 ....{-}2{--}
   1955 ..{--}
   1956 ..{- class="operator"}={--}
   1957 ..{- class="fraction"}
   1958 ....{-}1{- class="operator"}×{--}1{--}
   1959 ....{-}1{- class="operator"}×{--}2{--}
   1960 ..{--}
   1961 ..{- class="operator"}={--}
   1962 ..{- class="fraction"}
   1963 ....{-}1{--}
   1964 ....{-}2{--}
   1965 ..{--}
   1966 {--}
   1967 
   1968 Q.E.D.
   1969 
   1970 {- class="math block theorem"}
   1971 ..{- class="fraction"}
   1972 ....{- class="variable"}a{--}
   1973 ....{- class="variable"}b{--}
   1974 ..{--}
   1975 ..{- class="operator"}÷{--}
   1976 ..{- class="fraction"}
   1977 ....{- class="variable"}c{--}
   1978 ....{- class="variable"}d{--}
   1979 ..{--}
   1980 ..{- class="operator"}={--}
   1981 ..{- class="fraction"}
   1982 ....{- class="variable"}a{--}
   1983 ....{- class="variable"}b{--}
   1984 ..{--}
   1985 ..{- class="operator"}×{--}
   1986 ..{- class="fraction"}
   1987 ....{- class="variable"}d{--}
   1988 ....{- class="variable"}c{--}
   1989 ..{--}
   1990 {--}