➿ Divisibility Rules Checker
Is 123456 divisible by 8? By 11? Every rule from 2 to 13 applied with its working shown — digit sums, last-digits checks, the truncate-and-double trick for 7 — so you learn the rule while getting the answer.
| ÷ | Verdict | Rule | Working |
|---|---|---|---|
| 2 | ✅ | last digit even | last digit 6 is even ✓ |
| 3 | ✅ | digit sum divisible by 3 | digit sum 1+2+3+4+5+6 = 21; 21 is a multiple of 3 |
| 4 | ✅ | last two digits divisible by 4 | last two digits 56 ÷ 4 divides exactly |
| 5 | — | ends in 0 or 5 | last digit is 6 |
| 6 | ✅ | divisible by both 2 and 3 | 2: ✓, 3: ✓ |
| 7 | — | truncate & subtract 2× last digit, repeat | 123456 → 12333 → 1227 → 108 → 6; 6 is not a multiple of 7 |
| 8 | ✅ | last three digits divisible by 8 | last three digits 456 ÷ 8 divides exactly |
| 9 | — | digit sum divisible by 9 | digit sum = 21; 21 is not a multiple of 9 |
| 10 | — | ends in 0 | last digit is 6 |
| 11 | — | alternating digit sum divisible by 11 | alternating sum (from the right) = 3; 3 is not a multiple of 11 |
| 12 | ✅ | divisible by both 3 and 4 | 3: ✓, 4: ✓ |
| 13 | — | truncate & add 4× last digit, repeat | 123456 → 12369 → 1272 → 135 → 33; 33 is not a multiple of 13 |
Each verdict is exact BigInt arithmetic — the "working" column shows the classroom rule applied so you can do it by hand next time. The digital root (repeated digit sum) is what the 3 and 9 rules are really computing. Runs locally.
How the divisibility rules checker works
For each divisor 2 through 13 the tool gives the exact verdict (BigInt arithmetic, up to 30-digit numbers) and then shows the classroom rule doing its work: last-digit checks for 2, 5 and 10; last-two and last-three digits for 4 and 8; digit sums for 3 and 9; the alternating digit sum for 11; combination rules for 6 and 12; and the truncation tricks for the awkward ones — remove the last digit and subtract twice it for 7, or add four times it for 13, repeating until the number is small enough to eyeball. The digital root (repeated digit summing until one digit remains) is computed alongside, since it's what the 3 and 9 rules are secretly calculating.
Divisibility rules are arithmetic's best party trick with a real payoff: they're how you simplify fractions mentally, spot factors before long division, and check answers. The rules for 7 and 13 are the ones nobody remembers — seeing them run step by step on your own number is the fastest way to finally internalize them.
Frequently asked questions
What are the divisibility rules?
The classics: 2 — last digit even; 3 — digit sum divisible by 3; 4 — last two digits; 5 — ends in 0/5; 6 — both 2 and 3; 8 — last three digits; 9 — digit sum; 10 — ends in 0; 11 — alternating digit sum; 7 and 13 — truncation tricks the tool demonstrates.
How does the rule for 7 work?
Chop off the last digit, double it, subtract from what remains; repeat. 826 → 82 − 12 = 70, and 70 is a multiple of 7, so 826 is too. The tool runs this chain on your number and shows every step.
What is a digital root?
Keep summing the digits until one digit remains: 123456 → 21 → 3. It equals the number mod 9 (with 9 in place of 0) — which is why it powers the 3 and 9 rules and the old "casting out nines" check.
Why does the digit-sum rule work?
Because 10 ≡ 1 (mod 3 and mod 9), every power of ten leaves remainder 1 — so a number leaves the same remainder as its digit sum. The alternating-sum rule for 11 works the same way with 10 ≡ −1 (mod 11).
Is there a size limit?
Numbers up to 30 digits, checked exactly with BigInt — the rules' working stays readable at that size. Runs locally.