Test divisibility of any number by 2–12 and custom divisors. See rules, digit sum method, remainder, comprehensive rules table, visual divisibility grid, and presets.
The **Divisibility Test Calculator** instantly checks whether a number is divisible by common divisors 2 through 12 — plus any custom divisor you choose — and shows the specific rule that applies in each case. Instead of performing long division, you can verify divisibility at a glance using time-tested shortcuts that have been taught in classrooms for centuries.
Each divisor has its own rule. Divisibility by 2 checks the last digit; by 3, the digit sum; by 4, the last two digits; by 5, the last digit; by 6, both rules for 2 and 3; by 7, a doubling-and-subtraction method; by 8, the last three digits; by 9, the digit sum; by 10, the final zero; by 11, the alternating digit sum; and by 12, the rules for both 3 and 4. This calculator applies each rule automatically and shows the intermediate values so you can learn and verify the rules yourself.
The **digit sum method** is highlighted prominently — it is the foundation of divisibility tests for 3 and 9, and understanding it deepens your grasp of modular arithmetic. The calculator also shows the exact remainder for each divisor, so even when a number is not divisible, you know how far off it is.
A comprehensive divisibility rules reference table documents every rule from 2 to 12 with examples. The visual divisibility grid gives you a color-coded overview showing which divisors work for the given number. Preset buttons load interesting numbers like 2520 (the smallest number divisible by 1–10), 360, 7919 (a prime), and others. This is an essential tool for students learning arithmetic shortcuts, teachers planning lessons, and anyone who wants to check divisibility without a calculator.
Memorising and applying the digit-sum rule for 3, alternating-sum rule for 11, and the double-and-subtract rule for 7 across a large number is time-consuming and mistake-prone. This calculator tests divisibility by every standard divisor (2–12 and any custom divisor) in one click, shows which rule was applied, and gives the exact remainder. Students practising number theory, teachers preparing quiz keys, and puzzlers checking LCM-related claims all get instant, verifiable answers.
Divisibility by d: n is divisible by d if and only if n mod d = 0. Digit sum rule (for 3, 9): sum all digits; if the sum is divisible, so is the number.
Result: 2+3+4+0 = 9
Is 2,340 divisible by 9? Digit sum = 2+3+4+0 = 9. Since 9 is divisible by 9, yes — 2,340 ÷ 9 = 260.
Divisibility rules are shortcuts that let you determine whether one integer divides another without performing full long division. They exploit properties of modular arithmetic. For instance, because 10 ≡ 1 (mod 9), every power of 10 also ≡ 1, so a number and its digit sum leave the same remainder when divided by 9 (or by 3). Similarly, 10 ≡ −1 (mod 11), which means alternating powers of 10 have alternating signs — giving rise to the alternating-digit-sum rule for 11. The rule for 4 checks only the last two digits because 100 ≡ 0 (mod 4), so higher digits contribute nothing to the remainder.
For any divisor d, you can build a rule by examining 10^k mod d for successive k values. If the sequence is simple (e.g., constant or alternating), you get a digit-based shortcut. If not, the generic approach is to form telescoping sums or to use the "osculator" method: multiply the last digit by a fixed constant and add to or subtract from the truncated number. While these rules exist for every d, they become unwieldy for larger d; at that point, direct modular arithmetic (long division) is more practical. This calculator lets you test *any* divisor, custom or standard.
Divisibility is the gateway to deeper concepts: primes (numbers with exactly two divisors), GCD and LCM (built from shared and combined divisors), modular arithmetic (the foundation of cryptography), and Diophantine equations (integer-only solutions). Checking divisibility by small primes is the first step in trial-division factoring. Euler's totient function counts integers less than n that are coprime to n — itself based on divisibility relationships. Practically, divisibility checks appear in check-digit algorithms (ISBN, UPC, credit-card numbers via the Luhn algorithm), hash functions, and load-balancing schemes where tasks are assigned modulo the number of servers.
Add up all the digits. If the sum is divisible by 3, the original number is too. For 123: 1+2+3 = 6, and 6 ÷ 3 = 2, so 123 is divisible by 3.
Alternate adding and subtracting digits from right to left. If the result is divisible by 11, so is the number. For 2728: 8−2+7−2 = 11, divisible by 11 ✓.
Double the last digit and subtract it from the rest of the number. Repeat until you get a small number. Divisible by 7 at the end → original is too.
2520. It equals LCM(1,2,3,4,5,6,7,8,9,10). Try it with the 2520 preset!
Because 10 ≡ 1 (mod 3) and 10 ≡ 1 (mod 9), so any power of 10 contributes the same as its coefficient. The digit sum preserves the remainder mod 3 and mod 9.
Yes! Enter a custom divisor and the calculator will compute n mod d and show whether the remainder is zero.
Divisibility tests whether a specific d divides n. Factoring finds all prime factors of n. Divisibility is a quick yes/no check; factoring is more comprehensive.