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A collection of a builder and various scripts creating the noxz.tech sites
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commit df03c50ac9de961054842bed71521dc82edf7785
parent e4aba06600349e1e5c455a313bbed0d595859354
Author: Chris Noxz <chris@noxz.tech>
Date:   Mon, 23 Sep 2019 18:18:53 +0200

More divisibility

Diffstat:
Mnoxz.tech/guides/mathematics/index.md | 876++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++---
Mnoxz.tech/pub/style.css | 11+++++++++--
2 files changed, 853 insertions(+), 34 deletions(-)

diff --git a/noxz.tech/guides/mathematics/index.md b/noxz.tech/guides/mathematics/index.md @@ -9,7 +9,15 @@ out there. {:: class="toc"} {- class="toc-title"}Contents{--} + 1 [Divisibility rules](#math-div-rules) - + 1.1 [Divisibility by 2](#math-div-rules-2) + + 1.1 [Divisibility by 1](#math-div-rules-1) + + 1.2 [Divisibility by 2](#math-div-rules-2) + + 1.3 [Divisibility by 3](#math-div-rules-3) + + 1.4 [Divisibility by 4](#math-div-rules-4) + + 1.5 [Divisibility by 5](#math-div-rules-5) + + 1.6 [Divisibility by 6](#math-div-rules-6) + + 1.7 [Divisibility by 8](#math-div-rules-8) + + 1.8 [Divisibility by 9](#math-div-rules-9) + + 1.9 [Divisibility by 10](#math-div-rules-10) + 2 [Fractions](#math-fractions) + 2.1 [The fraction flip when dividing](#math-fractions-flip) {::} @@ -18,82 +26,886 @@ out there. --------------------------------------------- When it comes to divisibility there exists some neat methods to test a certain number's different divisibilities, or factors. Following are those methods and -their proof. +their proof. Some of the proofs are more trivial than others, such as the proof +for divisibility by 1, 2, 5 and 10. + +### {- id="math-div-rules-1"}Divisibility by 1{--} +This rule is quite easy to remember. Every integer is divisible by 1. + +{- class="math block theorem"} +..{- class="expression"} +....{-}1{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}n{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{- class="variable"}n{--} +....{- class="operator"}&isin;{--} +....{- class="variable"}&#x2124;{--} +..{--} +{--} ### {- id="math-div-rules-2"}Divisibility by 2{--} It's common knowledge that every even number (numbers ending with an even number) is divisible by 2. This is because even numbers are multiples of 2. In short, if a number ends with either 0, 2, 4, 6 or 8 it is divisible by 2. -Say we have a three digit number -{- class="math"}{- class="variable"}xyz{--}{--}, where - {- class="math"}{- class="variable"}x{--}{--} represents the number of - hundreds, - {- class="math"}{- class="variable"}y{--}{--} the number of tens and - {- class="math"}{- class="variable"}z{--}{--} the number of ones. This number +Say we have a four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number + can be represented as +{- class="math"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--}. If {- class="math"}{- class="variable"}d{--}{--} is divisible by 2 we can + represent it as an even number +{- class="math"} +..{- class="expression"} +....{-}2{--} +....{- class="variable"}n{--} +..{--} +{--}, like so: + +{- class="math block"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{-}2{--} +....{- class="variable"}n{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}2{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}500{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}50{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{-}5{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}n{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}2{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}2{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}d{--} +..{--} +{--} + +We can now see that the number is divisible by 2 if, and only if, the last +digit is divisible by 2. And so the theorem is proven. + +### {- id="math-div-rules-3"}Divisibility by 3{--} +The rule goes that if the sum of all digits in a number is divisible by 3, the +whole number is divisible by 3, i.e. + {- class="math"} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--}. Why is this proposition true? + +Let's use the four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number + can be represented as +{- class="math"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--}. Now let's break out one a, b and c from the first 3 terms, and factor out + 3 like so: + +{- class="math block"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}999{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}99{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{-}9{--} +......{- class="variable"}c{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}3{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}333{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}33{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{-}3{--} +......{- class="variable"}c{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +We can now see that the first term is divisible by 3, and the second term is +divisible by 3 if, and only if, the sum + {- class="math"} +..{- class="expression"} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} is divisible by 3. And so the theorem is proven. + +### {- id="math-div-rules-4"}Divisibility by 4{--} +The rule goes that if the last two digits of a number is divisible by 4, the +whole number is divisible by 4, i.e. + {- class="math"} +..{- class="expression"} +....{-}4{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}4{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}cd{--} +..{--} +{--}. Why is this proposition true? + +Let's use the four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number + can be represented as +{- class="math"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--}. Now let's factor out 4 from the first two terms, like so: + +{- class="math block"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}4{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}250{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}25{--} +......{- class="variable"}b{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}10{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}4{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}4{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}cd{--} +....{--} +..{--} +{--} + +We can now see that the first term is divisible by 4, so the whole number is +divisible by 4 if, and only if, the second term is divisible by 4. And so the +theorem is proven. + +### {- id="math-div-rules-5"}Divisibility by 5{--} +The rule goes that if the last digits of a number is divisible by 5, the +whole number is divisible by 4, i.e. + {- class="math"} +..{- class="expression"} +....{-}5{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}5{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}d{--} +..{--} +{--}. Why is this proposition true? + +Let's use the four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number + can be represented as +{- class="math"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--}. Now let's factor out 5 from the first three terms, like so: + +{- class="math block"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}5{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}500{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}50{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{-}5{--} +......{- class="variable"}c{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}5{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}5{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}d{--} +....{--} +..{--} +{--} + +We can now see that the first term is divisible by 5, so the whole number is +divisible by 5 if, and only if, the second term is divisible by 5. And so the +theorem is proven. As the only one digit numbers that are divisible by 5 are 0 +and 5, another way of putting it is -- if last digit is 0 or 5, the number is +divisible by 5. + +### {- id="math-div-rules-6"}Divisibility by 6{--} +This rule is a combination of the rule for [*divisibility by +2*](#math-div-rules-2) and [*divisibility by 3*](#math-div-rules-3). + +{- class="math block theorem"} +..{- class="expression"} +....{-}6{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +..{- class="operator"}&and;{--} +..{- class="expression"} +....{-}2{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}d{--} +..{--} +{--} + +### {- id="math-div-rules-8"}Divisibility by 8{--} +The rule is quite similar to the rule for *divisibility by 4*. The rule goes +that if the last three digits of a number is divisible by 8, the whole number +is divisible by 8, i.e. + {- class="math"} +..{- class="expression"} +....{-}8{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}8{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}bcd{--} +..{--} +{--}. Why is this proposition true? + +Let's use the four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number can be represented as {- class="math"} ..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--}. Now let's factor out 8 from the first three terms, like so: + +{- class="math block"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} ....{-}100{--} -....{- class="variable"}x{--} +....{- class="variable"}b{--} ....{- class="operator"}+{--} ....{-}10{--} -....{- class="variable"}y{--} +....{- class="variable"}c{--} ....{- class="operator"}+{--} -....{- class="variable"}z{--} +....{- class="variable"}d{--} ..{--} -{--}. If {- class="math"}{- class="variable"}z{--}{--} is divisible by 2 we can - represent it as an even number +{--} + +{- class="math block"} +..{- class="expression"} +....{-}8{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}125{--} +......{- class="variable"}a{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}100{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{-}10{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}8{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}8{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}bcd{--} +....{--} +..{--} +{--} + +We can now see that the first term is divisible by 8, so the whole number is +divisible by 8 if, and only if, the second term is divisible by 8. And so the +theorem is proven. + +### {- id="math-div-rules-9"}Divisibility by 9{--} +Much like the rule for *divisibility by 3*, the rule goes that if the sum of +all digits in a number is divisible by 9, the whole number is divisible by 9, +i.e. + {- class="math"} +..{- class="expression"} +....{-}9{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}9{--} +....{- class="operator"}&#x2223;{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--}. Why is this proposition true? + +Let's use the four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number + can be represented as {- class="math"} ..{- class="expression"} -....{-}2{--} -....{- class="variable"}n{--} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} ..{--} -{--}, like so: +{--}. Now let's break out one a, b and c from the first 9 terms, and factor out + 9 like so: {- class="math block"} ..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} ....{-}100{--} -....{- class="variable"}x{--} +....{- class="variable"}b{--} ....{- class="operator"}+{--} ....{-}10{--} -....{- class="variable"}y{--} +....{- class="variable"}c{--} ....{- class="operator"}+{--} -....{-}2{--} -....{- class="variable"}n{--} +....{- class="variable"}d{--} ..{--} -..{- class="operator"}={--} +{--} + +{- class="math block"} ..{- class="expression"} -....{-}2{--} ....{- class="fenced parenthesis"} ......{-}({--} -......{-}50{--} -......{- class="variable"}x{--} +......{-}999{--} +......{- class="variable"}a{--} ......{- class="operator"}+{--} -......{-}5{--} -......{- class="variable"}y{--} +......{-}99{--} +......{- class="variable"}b{--} ......{- class="operator"}+{--} -......{- class="variable"}n{--} +......{-}9{--} +......{- class="variable"}c{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} ......{-}){--} ....{--} ..{--} {--} {- class="math block"} +..{- class="expression"} +....{-}9{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}111{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}11{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{-}{--} +......{- class="variable"}c{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} ..{- class="operator"}&rArr;{--} ..{- class="expression"} -....{-}2{--} +....{-}9{--} ....{- class="operator"}&#x2223;{--} -....{- class="variable"}xyz{--} +....{- class="variable"}abcd{--} ..{--} ..{- class="operator"}&hArr;{--} ..{- class="expression"} -....{-}2{--} +....{-}9{--} +....{- class="operator"}&#x2223;{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +We can now see that the first term is divisible by 9, and the second term is +divisible by 9 if, and only if, the sum + {- class="math"} +..{- class="expression"} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} is divisible by 9. And so the theorem is proven. + +### {- id="math-div-rules-10"}Divisibility by 10{--} +The rule goes that if the last digits of a number is divisible by 10, the +whole number is divisible by 10, i.e. + {- class="math"} +..{- class="expression"} +....{-}10{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}10{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}d{--} +..{--} +{--}. Why is this proposition true? + +Let's use the four digit number +{- class="math"}{- class="variable"}abcd{--}{--}, where + {- class="math"}{- class="variable"}a{--}{--} represents the number of + thousands, + {- class="math"}{- class="variable"}b{--}{--} the number of hundreds, + {- class="math"}{- class="variable"}c{--}{--} the number of tens and + {- class="math"}{- class="variable"}d{--}{--} the number of ones. This number + can be represented as +{- class="math"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--}. Now let's factor out 10 from the first three terms, like so: + +{- class="math block"} +..{- class="expression"} +....{-}1000{--} +....{- class="variable"}a{--} +....{- class="operator"}+{--} +....{-}100{--} +....{- class="variable"}b{--} +....{- class="operator"}+{--} +....{-}10{--} +....{- class="variable"}c{--} +....{- class="operator"}+{--} +....{- class="variable"}d{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}10{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{-}100{--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{-}10{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{-}){--} +....{--} +....{- class="operator"}+{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +{--} + +{- class="math block theorem"} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}10{--} +....{- class="operator"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}10{--} ....{- class="operator"}&#x2223;{--} -....{- class="variable"}z{--} +....{- class="variable"}d{--} +....{--} ..{--} {--} +We can now see that the first term is divisible by 10, so the whole number is +divisible by 10 if, and only if, the second term is divisible by 10. And so the +theorem is proven. As the only one digit number that is divisible by 10 is 0, +another way of putting it is -- if last digit is 0, the number is +divisible by 10. + {- id="math-fractions"}Fractions{--} ------------------------------------ @@ -476,7 +1288,7 @@ know this should result in one half. ..{--} {--} -{- class="math block"} +{- class="math block theorem"} ..{- class="operator"}&rArr;{--} ..{- class="fraction"} ....{- class="variable"}a{--} diff --git a/noxz.tech/pub/style.css b/noxz.tech/pub/style.css @@ -258,8 +258,15 @@ ul.repo-log li .log-date { } .math.block { - display : block; - padding-left : 2em; + display : inline-block; + margin-left : 2em; +} + +.math.theorem { + padding : 1em 1.5em; + border : 1px double #000; + outline : 2px solid #000; + outline-offset : -4px; } .math .hi,