Divisibility

Description

Divisibility is the ability of a number to be evenly divided by another number. For example, four divided by two is equal to two, and therefore, four is divisible by two.

Notation

We commonly write $n|k$ . This means that n is a divisor of k. So for the example above, we would write 2|4.

Rules for common divisors

By $2^n$

A number is divisible by $2^n$ if the last ${n}$ digits of the number are divisible by $2^n$ .

By 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

By $5^n$

A number is divisible by $5^n$ if the last n digits are divisible by that power of 5.

By 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

By 7

Rule 1: Partition $n$ into 3 digit numbers from the right ( $d_3d_2d_1,d_6d_5d_4,\dots$ ). If the alternating sum ( $d_3d_2d_1 - d_6d_5d_4 + d_9d_8d_7 - \dots$ ) is divisible by 7 then the number is divisible by 7.

Rule 2: Truncate the last digit of ${n}$ , and subtract twice that digit from the remaining number. If the result is divisible by 7, then the number is divisible by 7. This process can be repeated for large numbers.

By 11

A number is divisible by 11 if the alternating sum of the digits is divisible by 11.

By 13

See rule 1 for divisibility by 7, a number is divisible by 13 if the same specified sum is divisible by 13.

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