You can tell whether a number divides evenly by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 — without doing a single division — just by checking its digits.
Divisibility rules are shortcuts — quick digit checks that instantly reveal whether one number divides evenly into another, with zero remainder, before you ever pick up a calculator or do long division.
A number is divisible by another number if dividing them leaves no remainder at all. 12 is divisible by 3, because 12 ÷ 3 = 4 exactly. 13 is not divisible by 3, because 13 ÷ 3 leaves a remainder of 1.
Rule: the last digit is 0, 2, 4, 6, or 8 (in other words, the number is even).
246: the last digit is 6, which is even, so 246 is divisible by 2 (246 ÷ 2 = 123).
Rule: add up all the digits. If that sum is divisible by 3, so is the original number.
4536: digits add up to 4+5+3+6 = 18. Since 18 is divisible by 3, so is 4536 (4536 ÷ 3 = 1512).
Rule: look only at the last two digits. If that two-digit number is divisible by 4, the whole number is too.
1728: the last two digits form 28, and 28 ÷ 4 = 7 exactly. So 1728 is divisible by 4 (1728 ÷ 4 = 432).
Rule: the last digit is 0 or 5.
3125: the last digit is 5, so 3125 is divisible by 5 (3125 ÷ 5 = 625).
Rule: the number must pass both the rule for 2 and the rule for 3 at the same time — divisible by 6 simply means divisible by both of its factors, 2 and 3.
4536: it's even (passes rule of 2) and its digit sum 18 is divisible by 3 (passes rule of 3). Since both pass, 4536 is divisible by 6 (4536 ÷ 6 = 756).
Rule: this one takes a few more steps. Double the last digit, subtract that from the rest of the number, and repeat until you reach a small number you can check directly. If the final result is divisible by 7, so is the original number.
812: double the last digit (2×2=4). Subtract from the rest: 81 − 4 = 77. Since 77 ÷ 7 = 11 exactly, 812 is divisible by 7 (812 ÷ 7 = 116).
The rule for 7 is trickier than the others because 7 doesn't divide evenly into 10 or 100 the way 2, 5, and other factors of 10 do — so no simple "look at the last digit" shortcut exists. Many students simply divide by 7 directly instead of using this rule, and that's a perfectly reasonable choice.
Rule: look only at the last three digits. If that three-digit number is divisible by 8, the whole number is too.
1728: the last three digits form 728, and 728 ÷ 8 = 91 exactly. So 1728 is divisible by 8 (1728 ÷ 8 = 216).
Rule: add up all the digits. If that sum is divisible by 9, so is the original number — the same idea as the rule for 3, just checking a stricter total.
4536: digits add up to 4+5+3+6 = 18, and 18 is divisible by 9. So 4536 is divisible by 9 (4536 ÷ 9 = 504).
Rule: the last digit is 0.
8710: the last digit is 0, so 8710 is divisible by 10 (8710 ÷ 10 = 871).
Rule: starting from the right, alternately add and subtract each digit. If the result is 0 or a multiple of 11, the original number is divisible by 11.
8712: from the right, alternate adding and subtracting: 2 − 1 + 7 − 8 = 0. Since 0 counts as a multiple of 11, 8712 is divisible by 11 (8712 ÷ 11 = 792).
Rule: the number must pass both the rule for 3 and the rule for 4 — just like 6 needed both 2 and 3, since 12 = 3 × 4.
4536: digit sum 18 is divisible by 3, and the last two digits (36) are divisible by 4. Since both pass, 4536 is divisible by 12 (4536 ÷ 12 = 378).
| Divisible By | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Digit sum is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Passes both the rule for 2 and the rule for 3 |
| 7 | Double the last digit, subtract from the rest, repeat |
| 8 | Last three digits form a number divisible by 8 |
| 9 | Digit sum is divisible by 9 |
| 10 | Last digit is 0 |
| 11 | Alternating digit sum (right to left) is 0 or a multiple of 11 |
| 12 | Passes both the rule for 3 and the rule for 4 |
Divisibility by 3 doesn't guarantee divisibility by 9 — 9 is a stricter check.
The fix: 96432 has a digit sum of 24. Since 24 is divisible by 3, the number passes the rule for 3. But 24 is not divisible by 9 (24 ÷ 9 = 2 remainder 6), so 96432 fails the rule for 9, even though it passed for 3. Always check the specific rule for the number you actually need — passing a smaller divisor's rule doesn't guarantee passing a larger one's.
Tap any number below to instantly see every number from 2 to 12 it divides evenly by.
Splitting a restaurant bill of $84 evenly among a group: knowing 84 is divisible by 2, 3, 4, 6, 7, and 12 (but not 5, 8, or 9) instantly tells you which group sizes divide the bill evenly, without a single division on a calculator.
The rule for 9 works for a surprising reason: since our number system is base 10, and 10 leaves a remainder of 1 when divided by 9, every place value (1, 10, 100, 1000...) also leaves a remainder of 1 when divided by 9 — which is exactly why simply adding the digits works.
Tap the correct answer for each question. Your score is tracked as you go, and your best score is saved on this device.
Divisibility rules are the foundation for everything that comes next in number theory — factors, multiples, simplifying fractions, and finding the HCF and LCM of numbers all lean on being able to spot divisibility quickly. They also make everyday splitting, grouping, and sharing problems fast to solve in your head.
12 questions. Select an answer for each, then submit to see your score instantly.