Lesson OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11

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<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 lessons in this site on divisibility numbers are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A>

&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>

Below are short annotations to the lessons with problems.

<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;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1 &nbsp;In the number 12345_67 one digit was omitted, &nbsp;and you see the blank placeholder in the corresponding position.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Restore the digit in the number in a way that the number is divisible by &nbsp;9.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;In the number 12345_678 one digit was omitted, &nbsp;and you see the blank placeholder in the corresponding position.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Is the solution unique? &nbsp;How many solutions are there?

<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;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1 &nbsp;In the number 123456_87 one digit was omitted, &nbsp;and you see the blank placeholder in the corresponding position.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2 &nbsp;In the number 12349_92 one digit was omitted, &nbsp;and you see the blank placeholder in the corresponding position.

<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;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1 &nbsp;Can there be a perfect square whose digits consist of exactly &nbsp;4&nbsp; ones, &nbsp;4&nbsp; twos and &nbsp;4&nbsp; zeros in any order?

<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;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1 &nbsp;Find the number of &nbsp;7-digit positive integers, &nbsp;where the sum of the digits is divisible by &nbsp;3.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2 &nbsp;Find the number of &nbsp;7-digit numbers, &nbsp;where the sum of the digits is divisible by &nbsp;9.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3 &nbsp;A five digit number is the fourth power of an integer. &nbsp;The sum of the first, &nbsp;third, &nbsp;and fifth digits

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;equals the sum of the second and fourth digits. &nbsp;What digit is in the thousands place ?

<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;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1 &nbsp;Here is a multiplication involving two &nbsp;16-digit numbers:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3851902343886132 * 5221791683705111

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The &nbsp;32-digit product is &nbsp;&nbsp;20 113 831 625 748 8_8 690 240 050 420 652.

<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;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1 &nbsp;Find the remainder when &nbsp;2&nbsp; raised to &nbsp;512 degree is divided by &nbsp;14.

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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