Divisibility Test Calculator
Enter any integer and choose a divisor to check divisibility in one click. The calculator applies the classic shortcut rule for that divisor (digit sums, last-digit tests, alternating block sums) and shows every step so you can follow along. Switch to Summary mode to see all twelve divisors from 2 to 13 tested at once.
What is a divisibility test?
A divisibility test is a shortcut rule that tells you whether one integer divides evenly into another without actually performing the full division. Instead of computing a quotient and checking the remainder, you examine just a few digits of the number according to the rule for your chosen divisor. These shortcuts work because our base-10 number system has predictable patterns: multiplying a number by 10 just shifts every digit one place left, so residues with respect to divisors that share factors with 10 (like 2 and 5) depend only on the final digit, while divisors related to 9 (like 3 and 9 themselves) depend on the digit sum.
How divisibility rules work
Every integer can be written as a sum of its digits multiplied by powers of 10. For divisor 3, because 10 leaves a remainder of 1 when divided by 3, each power of 10 also leaves a remainder of 1. That means the whole number leaves the same remainder as the sum of its digits. The rule for 9 works for the same reason. For 2, 4, and 8, because 10, 100, and 1000 are all divisible by 2, 4, and 8 respectively, only the last 1, 2, or 3 digits matter. For 11, the alternating sign arises because 10 leaves a remainder of -1 mod 11, so powers of 10 alternate between +1 and -1. The rules for 7 and 13 use blocks of three digits because 1000 is congruent to -1 mod 7 and also mod 13, so blocks of three digits alternate in sign.
Composite divisors and combined rules
Some divisors have no simple single-step rule, but can be tested by combining rules for their prime factors. Divisibility by 6 requires divisibility by both 2 and 3. Divisibility by 12 requires divisibility by both 3 and 4 (or equivalently by 4 and 3, which are coprime). When two divisors share no common factor, you can always combine their tests this way: a number is divisible by their product if and only if it is divisible by each of them separately.
What is a highly composite number?
A highly composite number is a positive integer that has more divisors than any smaller positive integer. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520 ... The number 2520 is notable as the smallest positive integer divisible by all integers from 1 through 10. Numbers that are divisible by many of the common divisors 2 through 13 tend to appear as the denominators of fractions in everyday measurement systems: 360 degrees in a circle, 12 inches in a foot, 60 minutes in an hour, and 24 hours in a day are all chosen because they divide evenly by many small integers.
Divisibility rules quick reference
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 (even). | 128 - last digit 8 is even. YES |
| 3 | Sum of all digits is divisible by 3. | 123 - 1+2+3=6, 6/3=2. YES |
| 4 | Last two digits form a number divisible by 4. | 1312 - 12/4=3. YES |
| 5 | Last digit is 0 or 5. | 225 - last digit 5. YES |
| 6 | Divisible by both 2 and 3. | 132 - even, digit sum 6. YES |
| 7 | Alternating sum of 3-digit blocks (right to left) is divisible by 7. | 1001 - 1-001=-0... check: 1001/7=143. YES |
| 8 | Last three digits form a number divisible by 8. | 1512 - 512/8=64. YES |
| 9 | Sum of all digits is divisible by 9. | 729 - 7+2+9=18, 18/9=2. YES |
| 10 | Last digit is 0. | 4560 - ends in 0. YES |
| 11 | Alternating digit sum (right to left) is divisible by 11. | 121 - 1-2+1=0. YES |
| 12 | Divisible by both 3 and 4. | 144 - digit sum 9 (div 3), 44/4=11. YES |
| 13 | Alternating sum of 3-digit blocks is divisible by 13. | 2197 - 2-197=-195, 195/13=15. YES |
Classic shortcut rules for checking divisibility without a calculator.
Frequently asked questions
What is the divisibility rule for 7?
For 7, take the alternating sum of 3-digit blocks of the number reading from right to left, and check whether that sum is divisible by 7. For example, for 1,001,001 you have 1 - 001 + 001 = 1, which is not divisible by 7 (and indeed 1001001 / 7 = 143000.14..., not exact). Another way: repeatedly double the last digit, subtract it from the rest, and repeat. This works because 1000 is congruent to -1 modulo 7.
Why does the digit-sum rule work for 3 and 9?
Because 10 is congruent to 1 modulo 3 (and also modulo 9), every power of 10 is also congruent to 1 modulo 3 and 9. When you expand a number like 345 as 3x100 + 4x10 + 5, each power of 10 contributes just 1 to the remainder, so the total remainder equals the sum of the digits. If that digit sum is divisible by 3 (or 9), so is the original number.
How do I test divisibility by a number not in the list, like 14?
Factorize the divisor into coprime parts and test each separately. For 14, note that 14 = 2 x 7 and gcd(2,7) = 1, so a number is divisible by 14 if and only if it is divisible by both 2 and 7. Test divisibility by 2 (last digit even) and by 7 (alternating 3-digit block sum), and if both pass, the number is divisible by 14.
What is the difference between a remainder and a quotient?
When you divide integer A by integer B, the quotient is how many complete times B fits into A, and the remainder is what is left over. For example, 17 / 5 gives quotient 3 (since 5 fits three complete times into 17 giving 15) and remainder 2 (since 17 - 15 = 2). Divisibility means the remainder is exactly zero.
Is a number divisible by itself?
Yes, every non-zero integer is divisible by itself, with quotient 1 and remainder 0. Every integer is also divisible by 1, since 1 divides any integer with no remainder. These are called the trivial divisors.
Can I test divisibility of negative numbers?
Yes. In standard integer arithmetic, -24 is divisible by 6 because -24 / 6 = -4 exactly. The divisibility rules apply to the absolute value of the number; the sign of the number does not affect whether a remainder of zero occurs.