Parent guide · 6 min read
Divisibility rules for kids: the quick checks for 2, 3, 5, 7 and 9
A divisibility rule is a short test that tells you whether one number divides another exactly, without doing the division. "Does 3 go into 4,521?" can be answered in about five seconds by adding digits, and the same habit makes simplifying fractions, finding factors and checking long division much quicker. Children who only meet the rules as a list to memorise tend to mix them up, so this guide explains why the common ones work and gives a worked example to copy.
Try this tonight
5 things you can do at home
Warm up with the last-digit rules
Write ten random numbers up to 200 and ask which are divisible by 2, 5 and 10. Even numbers end in 0, 2, 4, 6 or 8. Multiples of 5 end in 0 or 5. Multiples of 10 end in 0. These need no arithmetic at all, which makes them a confident first win.
Teach the digit-sum trick for 3 and 9
Take 4,521. Add the digits: 4 + 5 + 2 + 1 = 12. Twelve is in the 3 times table, so 4,521 divides by 3. Twelve is not in the 9 times table, so it does not divide by 9. Let your child check with a calculator once so they can see the rule is not magic.
Add the last-digits rules for 4 and 8
For 4, look only at the last two digits: 316 works because 16 divides by 4. For 8, look at the last three digits: 5,128 works because 128 ÷ 8 = 16. Children find it easier to remember "two digits for 4, three digits for 8" as a pair.
Save 7 for last and do it slowly
Cover the last digit, double it, and subtract it from what is left. For 343: cover the 3, double it to get 6, and 34 − 6 = 28. Twenty-eight is in the 7 times table, so 343 divides by 7. If the answer is still big, repeat the same steps on it.
Finish by playing "which numbers can I use?"
Pick a number such as 360 and ask: "Which of 2, 3, 4, 5, 6, 8, 9 and 10 go into this?" Listing every rule that fits is a good check that none of them has been confused with another.
Why the digit-sum rule works
Each place value in a number is one more than a multiple of 9: 10 is 9 + 1, 100 is 99 + 1, 1,000 is 999 + 1. So 4,521 is really 4 lots of 999 plus 4, 5 lots of 99 plus 5, 2 lots of 9 plus 2, and 1. All the 9-based chunks divide by both 3 and 9, which leaves only the digit sum (12) to decide.
That is why the rule works for 3 and for 9 but not for other numbers. If a child asks "does adding the digits work for 4?", the honest answer is no: 4 has its own rule that looks at the last two digits, because 100 divides by 4 exactly and so only the end of the number matters.
A worked example: is 2,520 divisible by 2, 3, 5, 7 and 9?
Last digit first. 2,520 ends in 0, so it divides by 2, 5 and 10. Digit sum: 2 + 5 + 2 + 0 = 9. Nine is in both the 3 and the 9 times tables, so 2,520 divides by 3 and by 9.
Now 7. Cover the last digit (0), double it (still 0) and subtract from 252: 252 − 0 = 252. Repeat: cover the 2, double it to get 4, and 25 − 4 = 21. Twenty-one is in the 7 times table, so 2,520 divides by 7. Check: 2,520 ÷ 7 = 360. This number is divisible by all of 2, 3, 5, 7 and 9, which is why it turns up so often in maths puzzles.
The mix-ups that cost marks
The most common slip is using the digit-sum rule on the wrong number. A digit sum of 12 shows a multiple of 3, but not of 9. Another is thinking that a number ending in 5 is divisible by 10: only a final 0 does that.
The other trap is stopping the 7 check too early. After one round of "double and subtract" the answer is smaller but may not be obviously in the 7 times table. Keep going until it is a number you recognise, such as 0, 7, 14, 21 or 28. If the end result is not a multiple of 7, the original number is not either.
Signs it’s time for outside help
When home help has done what it can
- Cannot say whether a number is even or a multiple of 5 without dividing.
- Mixes up the digit-sum rule (3 and 9) with the last-digit rules (2, 4, 5, 8 and 10).
- Struggles to find factor pairs, so every factor list has gaps.
- Needs a calculator to check small divisions that the times tables should answer.
- Does long division by trial and error instead of estimating which numbers will fit.
Questions parents ask
FAQ
What age do children usually learn divisibility rules?
Usually between ages 9 and 12 (around US Grades 4 to 6, and Australian Years 5 and 6), alongside factors, multiples and simplifying fractions. The rules for 2, 5 and 10 come first; 7 is often left for later.
Is there a divisibility rule for 6?
Yes, and it is a combination: a number divides by 6 if it divides by both 2 and 3. So check that it is even and that its digits add up to a multiple of 3. The same idea gives 12 (divisible by 3 and 4) and 15 (divisible by 3 and 5).
Why does the rule for 7 work?
Take a number written as 10a + b, where b is the last digit. Subtracting twice b from a gives a − 2b, and 10a + b is divisible by 7 exactly when a − 2b is. It is hard to see without algebra, so most children just practise the steps and check with division.
Do divisibility rules replace learning the times tables?
No. The rules narrow down which numbers might fit, but you still need the times tables to recognise results such as 21 or 28. They are a speed-up for factors and fractions, not a shortcut around fluency.
Further reading
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