commit: df03c50ac9de961054842bed71521dc82edf7785
parent: e4aba06600349e1e5c455a313bbed0d595859354
author: Chris Noxz <chris@noxz.tech>
date: Mon, 23 Sep 2019 18:18:53 +0200
More divisibility
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"}∣{--}
+....{- class="variable"}n{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{- class="variable"}n{--}
+....{- class="operator"}∈{--}
+....{- class="variable"}ℤ{--}
+..{--}
+{--}
### {- 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"}⇒{--}
+..{- class="expression"}
+....{-}2{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}2{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
+....{- 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"}⇒{--}
+..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}4{--}
+....{- class="operator"}∣{--}
+....{- 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"}⇒{--}
+..{- class="expression"}
+....{-}4{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}4{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}5{--}
+....{- 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 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"}⇒{--}
+..{- class="expression"}
+....{-}5{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}5{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="variable"}a{--}
+......{- class="operator"}+{--}
+......{- class="variable"}b{--}
+......{- class="operator"}+{--}
+......{- class="variable"}c{--}
+......{- class="operator"}+{--}
+......{- class="variable"}d{--}
+......{-}){--}
+....{--}
+..{--}
+..{- class="operator"}∧{--}
+..{- class="expression"}
+....{-}2{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}8{--}
+....{- class="operator"}∣{--}
+....{- 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"}⇒{--}
+..{- class="expression"}
+....{-}8{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}8{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}9{--}
+....{- class="operator"}∣{--}
+....{- 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"}⇒{--}
..{- class="expression"}
-....{-}2{--}
+....{-}9{--}
....{- class="operator"}∣{--}
-....{- class="variable"}xyz{--}
+....{- class="variable"}abcd{--}
..{--}
..{- class="operator"}⇔{--}
..{- class="expression"}
-....{-}2{--}
+....{-}9{--}
+....{- class="operator"}∣{--}
+....{- 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"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}10{--}
+....{- 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 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"}⇒{--}
+..{- class="expression"}
+....{-}10{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}10{--}
....{- class="operator"}∣{--}
-....{- 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"}⇒{--}
..{- 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,
