Divisibility rules

These divisibility rules help determine when integers are divisible by particular other integers.

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 11

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

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 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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