Lesson Divisibility by 2 rule

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<H2>Divisibility by 2 rule</H2>

The &nbsp;"Divisibility by 2" rule&nbsp; is as follows:

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 &nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;2&nbsp; if and only if its last digit is divisible by &nbsp;2.&nbsp;&nbsp;&nbsp;&nbsp;

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In other words, &nbsp;for checking if the given integer number is divisible by &nbsp;2, &nbsp;make the following steps:

&nbsp;&nbsp;&nbsp;&nbsp;1. &nbsp;Take the last digit of the number while ignoring the rest.

&nbsp;&nbsp;&nbsp;&nbsp;2. &nbsp;Check if this last digit is divisible by &nbsp;2.

&nbsp;&nbsp;&nbsp;&nbsp;3. &nbsp;If the last digit is divisible by &nbsp;2, &nbsp;then the original number is divisible by &nbsp;2.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the last digit is not divisible by &nbsp;2, &nbsp;then the original number is not divisible by &nbsp;2.

Quite simple, &nbsp;isn't?

<H3>Example 1</H3>Check if the number &nbsp;358&nbsp; is divisible by &nbsp;2.

Solution

The last digit of this number is &nbsp;8.

It is divisible by &nbsp;2.

Hence, &nbsp;the original number &nbsp;358&nbsp; is divisible by &nbsp;2, &nbsp;in accordance with the &nbsp;"Divisibility by 2" rule.

You may check it by making the direct division: &nbsp;{{{358/2}}} = {{{179}}}. &nbsp;It shows that the number &nbsp;358&nbsp; is divisible by &nbsp;2.

The &nbsp;Divisibility rule&nbsp; allows you to get the same conclusion without making long calculations.

<H3>Example 2</H3>Check if the number &nbsp;371&nbsp; is divisible by &nbsp;2.

Solution

The last digit of this number is &nbsp;1.

It is not divisible by &nbsp;2.

Hence, &nbsp;the original number &nbsp;371&nbsp; is not divisible by &nbsp;2, &nbsp;in accordance with the &nbsp;"Divisibility by 2" rule.

For those who loves the proofs, &nbsp;here it is :)

<H3>The proof of the "divisibility by 2" rule</H3>

First, &nbsp;let us consider how the &nbsp;"divisibility by 2"&nbsp; rule can be proved for the concrete number &nbsp;358, &nbsp;for example.

We can write &nbsp;&nbsp;358 = 3*100 + 5*10 + 8.

The first two additives, &nbsp;3*100&nbsp; and &nbsp;5*10, &nbsp;have the common multiple &nbsp;10, &nbsp;which is divisible by &nbsp;2. &nbsp;Hence, &nbsp;the sum of these additives &nbsp;3*100 + 5*10&nbsp;  is divisible by &nbsp;2.

Therefore, &nbsp;the divisibility by &nbsp;2&nbsp; of our original number depends on and is determined solely by the last additive, &nbsp;which is the last digit of the number, &nbsp;i.e. &nbsp;8&nbsp; in our case.

For the general case, &nbsp;the proof follows the same arguments.

Let &nbsp;{{{a[n]}}}{{{a[n-1]}}} . . . {{{a[2]}}}{{{a[1]}}}{{{a[0]}}}&nbsp; be the decimal record of our integer number &nbsp;N.

Then &nbsp;N = {{{a[n]*10^n}}} + {{{a[n-1]*10^(n-1)}}} + . . . + {{{10^2*a[2]}}} + {{{10*a[1]}}} + {{{a[0]}}}.

The first &nbsp;n&nbsp; additives, &nbsp;{{{a[n]*10^n}}}, &nbsp;{{{a[n-1]*10^(n-1)}}}, . . . , &nbsp;{{{10^2*a[2]}}}&nbsp; and &nbsp;{{{10*a[1]}}}&nbsp; have the common multiple &nbsp;10, &nbsp;which is divisible by &nbsp;2.

Hence, &nbsp;the sum of these additives &nbsp;{{{a[n]*10^n}}} + {{{a[n-1]*10^(n-1)}}} + . . . + {{{10^2*a[2]}}} + {{{10*a[1]}}}&nbsp; is divisible by &nbsp;2.

Therefore, &nbsp;the divisibility by &nbsp;2&nbsp; of our original number depends on and is determined solely by the last additive, &nbsp;which is presented by the last digit of the number.

It is what has to be proved.

<H2>OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11</H2>

The rules for divisibility by 2, 3, 4, 5, 6, 9, 10, and 11 are

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&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;2&nbsp; if and only if its last digit is divisible by &nbsp;2.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;3&nbsp; if and only if the sum of its digits is divisible by &nbsp;3.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;4&nbsp; if and only if the number formed by its two last digits is divisible by &nbsp;4.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;5&nbsp; if and only if its last digit is &nbsp;5&nbsp; or &nbsp;0.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;6&nbsp; if and only if its last digit is even and the sum of the digits is divisible by &nbsp;3.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;9&nbsp; if and only if the sum of its digits is divisible by &nbsp;9.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;10&nbsp; if and only if its last digit is &nbsp;0.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;11&nbsp; if and only if the alternate sum of its digits is divisible by &nbsp;11.&nbsp;&nbsp;&nbsp;&nbsp;

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(The &nbsp;alternate sum&nbsp; is the algebraic sum of digits where the sign is changed to the opposite from each digit to the next one).

My other lessons in this site on divisibility numbers are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-4-rule.lesson>Divisibility by 4 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-5-rule.lesson>Divisibility by 5 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-6-rule.lesson>Divisibility by 6 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-9-rule.lesson>Divisibility by 9 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-10-rule.lesson>Divisibility by 10 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-11-rule.lesson>Divisibility by 11 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-9.lesson>Restore the omitted digit in a number in a way that the number is divisible by 9</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-11.lesson>Restore the omitted digit in a number in a way that the number is divisible by 11</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Can-there-be-a-perfect-square.lesson>Can there be a perfect square ?</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problems-on-divisibility-numbers.lesson>Math circle level problems on divisibility numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-restoring-digit-in-the-product-of-two-16-digit-numbers.lesson>Math circle level problem on restoring digit in the product of two 16-digit numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-finding-remainders.lesson>Math circle level problem on finding remainders</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/OVERVIEW-of-Divisibility-rules-by-2-3-4-5-6-9-10-11.lesson>OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11</A>

Lessons that are closely adjacent to these are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Product-of-two-consecutive-integers-is-divisible-by-2.lesson>Product of two consecutive integers is divisible by 2</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Product-of-three-consecutive-integers-is-divisible-by-6.lesson>Product of three consecutive integers is divisible by 6</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Problems-with-consecutive-odd-even-integers/Problems-dealing-with-the-product-of-two-consecutive-integers.lesson>Problems dealing with the product of two consecutive integers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Problems-with-consecutive-odd-even-integers/Problems-dealing-with-the-product-of-three-consecutive-integers.lesson>Problems dealing with the product of three consecutive integers</A>

When you learn the listed lessons on divisibility rules, &nbsp;you will be able to formulate and to establish many other simple and useful divisibility rules like these:

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&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;8&nbsp; if and only if the number formed by its three last digits is divisible by &nbsp;8.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;15&nbsp; if and only its last digit is &nbsp;5&nbsp; or &nbsp;0&nbsp; and the sum of its digits is divisible by &nbsp;3.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;20&nbsp; if and only if its two last digits are &nbsp;20, &nbsp;40, &nbsp;60, &nbsp;80&nbsp; or &nbsp;00.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;25&nbsp; if and only if its two last digits are &nbsp;25, &nbsp;50, &nbsp;75&nbsp; or &nbsp;00.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;50&nbsp; if and only if its two last digits are &nbsp;50&nbsp; or &nbsp;00.&nbsp;&nbsp;&nbsp;&nbsp;

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