Remainder Rules

Divisibility by 3

The remainder of $n$ when divided by $3$ is equal to the remainder of the sum of the digits of $n$ when divided by $3$.

Divisibility by 9

The remainder of $n$ when divided by $9$ is equal to the remainder of the sum of the digits of $n$ when divided by $9$.

Divisibility by 11

The remainder of $n$ when divided by $11$ is equal to the remainder of the alternating sum of its digits.

Example:

From right to left, consider the signs $+$ and $-$ alternately.

Divisibility by 2, 5, and 10

Only the last digit matters.

Divisibility by 4, 25, and 100

Only the last two digits matter.

Divisibility by 8, 125, and 1000

Only the last three digits matter.