Divisibility rules

Knowing when a number is divisible by another without carrying out the division is extremely useful for decomposing quickly. Here are the classic rules, all provable with the same idea: working modulo $d$.

Property — Quick divisibility rules

A natural number $n$ is divisible by:

$d$	Rule (on the decimal digits of $n$)
$2$	last digit even ($0,2,4,6,8$)
$3$	sum of the digits divisible by $3$
$4$	last two digits form a number divisible by $4$
$5$	last digit $0$ or $5$
$6$	divisible by $2$ and by $3$
$7$	subtracting twice the last digit from the number without its last digit gives a multiple of $7$ (recursive)
$8$	last three digits form a number divisible by $8$
$9$	sum of the digits divisible by $9$
$10$	last digit $0$
$11$	alternating difference of the digits (from the last) divisible by $11$

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Topics: Numbers and operations
Concepts: Divisibility rules · Divisibility
Skills: Decomposing