Divisibility Calculator

How to Use the Divisibility Calculator

To use the divisibility calculator, type the number you want to test, type the number you want to divide by, and read the verdict. The answer and the working appear together in 4 steps:

1. Enter the number in the first field. Whole numbers up to 30 digits are handled exactly, since the arithmetic runs in arbitrary-precision integers rather than floating point.
2. Enter the divisor in the second field. The headline answers that one question with a yes or a no, the remainder and the quotient.
3. Read the rules table in the working panel. Every divisibility test from 2 to 13 is listed with the rule, the arithmetic applied to your own digits, and a pass or fail.
4. Open the extended table for divisors from 14 to 1000, including the divisibility test of 25, 125, 250, 500 and 1000.

The divisibility test calculator reports 4 results for every number: the yes-or-no verdict against your chosen divisor, the exact remainder, the list of divisors from 2 to 13 that pass, and every divisor the number has up to a trillion. A number with exactly 2 divisors is flagged as prime.

What Does Divisible Mean?

Divisible means the division leaves no remainder. A whole number n is divisible by a whole number k when n ÷ k comes out exactly, with a remainder of 0 and nothing left over.

18 is divisible by 3, since 18 ÷ 3 = 6 with no remainder. 18 is divisible by 1, 2, 6, 9 and 18 for the same reason. 18 is not divisible by 7, since 18 ÷ 7 = 2 with 4 left over.

Divisibility links 3 words that describe the same relationship from different sides. 3 is a divisor of 18 and a factor of 18, and 18 is a multiple of 3. Mathematicians write the relationship as 3 | 18, read as "3 divides 18".

Divisibility rules are procedures that answer that question without performing the division. Each rule replaces the original problem with a smaller one — a digit, a pair of digits, or a sum — that gives the same answer. Testing whether the 30-digit number 123456789012345678901234567890 is divisible by 3 takes one digit sum, while dividing it by hand takes 30 rounds of long division.

There are 6 strategies behind every divisibility rule, and each one shrinks the number in a different way:

Divisibility Calculator Example

Testing 918082 for divisibility by 11 takes 1 alternating sum and returns yes. Assign signs to the digits from the right, starting with a plus, then add them up:

22 is a multiple of 11, so 918082 is divisible by 11. The minus sign carries no weight in the test — an alternating sum of −22 and one of 22 both settle the question the same way. The calculator confirms it with the division: 918082 ÷ 11 = 83462 with a remainder of 0.

Testing 343 for divisibility by 7 uses a different strategy, and takes 2 rounds. Drop the last digit, double it, subtract it from what is left, and repeat until the answer is small enough to read:

28 is a multiple of 7, so 343 is divisible by 7. The division agrees: 343 ÷ 7 = 49 with a remainder of 0.

A failing case looks the same until the last line. Testing 9973 for divisibility by 7 gives 997 − 2×3 = 991, then 99 − 2×1 = 97, and 97 is not a multiple of 7. 9973 is not divisible by 7, which the division confirms as 9973 ÷ 7 = 1424 with a remainder of 5. 9973 fails every rule from 2 to 13, and the divisor list shows why — 9973 is prime.

Divisibility and Remainders

Divisibility is the single case where the remainder is 0. Every division of whole numbers produces a quotient and a remainder, and the remainder decides the verdict on its own.

The size of the remainder carries no partial credit. 96 ÷ 7 leaving 5 and 96 ÷ 5 leaving 1 are equally not divisible, since divisibility is a yes-or-no property rather than a measurement of how close the division came.

Dividing by k produces a remainder somewhere in the range 0 to k − 1, which gives k possible outcomes and only 1 of them meaning divisible. Dividing by 7 leaves 0, 1, 2, 3, 4, 5 or 6, so 1 remainder in 7 signals divisibility. Every divisibility rule works by preserving that remainder while shrinking the number: the digit sum of 729 leaves the same remainder modulo 9 as 729 itself.

Three tools answer three different questions about the same division. The remainder calculator returns the remainder as a number. The long division tableau tool draws every subtraction that produced it. The modulo operator calculator handles the sign conventions that appear once negative numbers enter.

Common Divisibility Rules

There are 12 divisibility rules in common use, covering the divisors 2 through 13. Each one reduces the number to something smaller that carries the same answer.

The rules for powers of 2, 5 and 10 follow one pattern. 1000 is a multiple of 8, so every digit above the hundreds column is already divisible by 8 and only the last 3 matter. The same argument applies at every power:

The rules for 3 and 9 come out of one line of algebra. Write a 5-digit number with digits a, b, c, d, e in the base-10 positional system, then split each power of ten into a multiple of 9 plus 1:

The right-hand side is a multiple of 9, and therefore a multiple of 3. The number n and the sum of its digits leave the same remainder when divided by 9, and the same remainder when divided by 3. Testing the digit sum answers the question about n exactly, for any number of digits.

The rule for 11 flips the sign. Ten leaves a remainder of −1 when divided by 11, a hundred leaves +1, a thousand leaves −1, so each column contributes its digit with an alternating sign. Adding and subtracting the digits from the right reproduces the remainder modulo 11 exactly.

The rules for 7 and 13 use blocks of 3 digits, and the reason sits in one product: 7 × 11 × 13 = 1001. A thousand leaves a remainder of −1 when divided by 7, by 11 and by 13, so an alternating sum of 3-digit blocks preserves the remainder for all 3 divisors at once. Testing 9111414 for 13 gives 414 − 111 + 9 = 312, and 312 = 13 × 24, so 9111414 is divisible by 13.

Every divisor above 13 is built from the rules already listed. Factor the divisor into prime powers, then test each prime power separately:

• 14 = 2 × 7 — test 2 and 7.
• 15 = 3 × 5 — test 3 and 5.
• 18 = 2 × 3² — test 2 and 9.
• 20 = 2² × 5 — test 4 and 5, which reduces to a last digit of 0 with an even digit before it.
• 24 = 2³ × 3 — test 8 and 3.
• 25 = 5² — last two digits are 00, 25, 50 or 75.
• 60 = 2² × 3 × 5 — test 4, 3 and 5.
• 125 = 5³ — last three digits divisible by 125, meaning 000, 125, 250, 375, 500, 625, 750 or 875.
• 250, 500 and 1000 — last three digits are 000, 250, 500 or 750 for 250; 000 or 500 for 500; 000 for 1000.

The prime powers have to be tested separately for a reason. Testing 12 as "divisible by 2 and by 6" fails, since 18 passes both and 18 ÷ 12 = 1 R 6. Splitting 12 into 4 × 3 works, since 4 and 3 share no prime factor. The extended table in the working panel applies all 20 of these composite and prime-power rules to your number at once.

Common Divisibility Mistakes

There are 5 common divisibility mistakes, and every one of them is caught by dividing and checking the remainder.

17352 → 52
52 ÷ 8 = 6 R 4  → "not divisible"

17352 → 352
352 ÷ 8 = 44 R 0  → divisible ✓

918082 → 9+1+8+0+8+2 = 28
28 ÷ 11 → "not divisible"

918082 → −9+1−8+0−8+2 = −22
22 = 11 × 2  → divisible ✓

12 = 2 × 6
18: even ✓, ÷6 ✓ → "divisible"
18 ÷ 12 = 1 R 6  ✗

12 = 4 × 3
18: last two 18 ÷ 4 ✗
→ not divisible ✓

343 → 343 − 2×3 = 337
337 ÷ 7 → "not divisible"

343 → 34 − 2×3 = 28
28 = 7 × 4  → divisible ✓

Check any divisibility answer in 1 division: divide and look at the remainder. A remainder of 0 confirms the rule, and any other remainder exposes the slip. The greatest common divisor of several numbers tool takes the next step once the small factors are known, and the extended Euclidean algorithm steps tool shows the divisions behind it.