Divisibility Rules | Moosa Academy

In a timed test there is rarely room for long division. A divisibility rule answers "does this divide exactly?" by glancing at a digit or two — and each rule is a genuine theorem, not a trick, as the last section explains.

Concept Rules that read the last digits

For these divisors you never look at the whole number — only its tail.

Divisor	Rule	Examples
2	The units digit is even:	, ,
4	The last two digits form a multiple of	(32), (16), (48)
5	The units digit is or	, ,
8	The last three digits form a multiple of	(120), (160), (024)
10	The units digit is	, ,

Concept Rules that add the digits

Divisor	Rule	Examples
3	The digit sum is a multiple of	, ,
9	The digit sum is a multiple of	, ,
6	Divisible by and by	, ,

The rule for shows a general shortcut: to test a composite divisor, split it into factors that share no common divisor and test each. Since , passing both tests is enough.

Theorem Dividing by 11

Add the digits in the odd positions, add those in the even positions, and subtract. If the difference is or a multiple of , the number is divisible by .

— divisible

Only the size of the difference matters, so counts exactly as does. Subtracting the other way round simply flips the sign.

Theorem Dividing by 7

Remove the units digit, double it, and subtract that from what remains. If the result is divisible by , so was the original number.

— divisible

This is the awkward one, and it is worth being honest about it: for a three-digit number, plain division by is usually quicker. Reach for the rule when the number is long, or skip it entirely.

Concept Why the rules work

Every rule comes from the remainder that leaves when divided by the number being tested.

3 and 9 —

leaves remainder

, so every place value counts the same. Adding the digits is enough.

11 —

leaves remainder

, so place values alternate in sign. Hence the alternating sum.

2, 4, 5, 8, 10 — powers of

become divisible by these from some point on, so only the final digits can affect the answer.

For , for instance, every power of from upward is already a multiple of . Whatever sits in the hundreds column and beyond is therefore irrelevant, leaving just the last two digits to check.

Summary

For , and , look only at the units digit.
For check the last two digits; for check the last three.
For and , add the digits and test the sum.
For , test and separately — both must pass.
For , take the alternating sum; or any multiple of works, sign ignored.
For , subtract twice the units digit from the rest — or just divide.
Each rule follows from the remainder leaves on division by that number.