MISSION · Number sense
Divisibility Rules
Quick tests that save you the long division
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
For the small divisors, a glance at the digits tells you whether the division will come out exactly — no working required.
The rules
| Divisible by | Test |
|---|---|
| 2 | Last digit is even |
| 3 | Digits add to a multiple of 3 |
| 4 | Last two digits make a multiple of 4 |
| 5 | Ends in 0 or 5 |
| 6 | Passes the tests for 2 and 3 |
| 8 | Last three digits make a multiple of 8 |
| 9 | Digits add to a multiple of 9 |
| 10 | Ends in 0 |
Seven has no test worth the trouble — just divide.
Why the digit-sum rules work
This is the one worth understanding rather than memorising, because it explains two rules at once.
Take 342. Write it out in full: \(300 + 40 + 2\), which is \(3 \times 99 + 3 + 4 \times 9 + 4 + 2\).
Every 9, 99, and 999 is already a multiple of 9 — and of 3. So all of that part divides cleanly whatever the digits are, and the only thing left to check is \(3 + 4 + 2 = 9\). That’s the digit sum, and it decides the whole question.
\(342\) has a digit sum of 9, so it’s divisible by both 3 and 9. And indeed \(342 = 9 \times 38\).
If the sum is still large, add again. For 98,765: \(9+8+7+6+5 = 35\), then \(3+5 = 8\) — not a multiple of 3, so neither is the original.
Why the 4 and 8 rules use the last digits
Because 100 is a multiple of 4, and 1000 is a multiple of 8. Everything above those columns divides cleanly on its own, so only the tail matters.
For 4, check the last two digits: 7,316 ends in 16, which is \(4 \times 4\), so the whole number divides by 4. For 8, check the last three.
Following the same logic, 2, 5, and 10 need only the last digit, because 10 is a multiple of all three.
What they’re for
Simplifying fractions. Faced with \(\frac{204}{558}\), both are even, and both digit sums (6 and 18) are multiples of 3 — so 6 divides both, and you’ve started without guessing.
Finding factors and primes. To test whether 221 is prime, the rules eliminate 2, 3, 5, and 10 in seconds, leaving only a few candidates to try properly. (It isn’t prime — \(13 \times 17\).)
Checking answers. If a problem’s answer must divide by 3 and yours doesn’t, something went wrong.
Common mix-ups
- Adding the digits for 4 or 8. Those rules use the last digits, not the sum.
- Checking only 2 for 6. It needs 3 as well: 10 is even but not divisible by 6.
- Assuming a rule for 7 exists in any useful form.
- Stopping at a large digit sum. Keep adding until it’s small enough to recognise.
Check yourself
Is 4,518 divisible by 3, by 6, and by 9?
Digit sum \(4+5+1+8 = 18\), a multiple of both 3 and 9 — so yes to both. It’s also even, so it passes 6 as well. Indeed \(4518 = 6 \times 753\).
LEVEL 3 · REFLECT
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