Divisibility Test Calculator

Enter a whole number and a divisor to instantly see whether it divides evenly, the exact quotient and remainder, and the digit-based shortcut rule behind the answer.

Quick Facts

Definition
N is divisible by D when N mod D = 0
Equivalently, N = D × q for some integer q, with no remainder.
Rule for 3 & 9
Check the digit sum
If the sum of a number's digits divides evenly by 3 (or 9), so does the number itself.
Rule for 2, 5 & 10
Check only the last digit
Even last digit → divisible by 2; last digit 0 or 5 → divisible by 5; last digit 0 → divisible by 10.
Rule for 11
Alternating digit sum
Add and subtract digits from right to left; a result divisible by 11 means the number is too.

Your Results

Calculated
Divisible?
-
Yes/No test result
Quotient
-
N ÷ D, integer part
Remainder
-
N mod D
Rule Applied
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Shortcut used for this divisor

Ready

Enter a whole number and a divisor, then press Calculate.

How the Divisibility Test Calculator Works

A whole number N is divisible by another whole number D when dividing N by D leaves a remainder of 0 — in other words, when N = D × q for some integer q. This calculator applies the division algorithm directly using exact integer arithmetic (so it stays accurate even for very large numbers) and also shows the digit-based shortcut rule that students and mathematicians use to check divisibility of common divisors, from 2 through 13, without doing long division.

The division algorithm behind every divisibility test

Every whole-number division can be written as N = D × q + r, where q is the quotient and r is the remainder, with 0 ≤ r < D. N is divisible by D exactly when r = 0. This calculator computes q and r directly from your inputs, so the divisibility result is always exact — never an approximation.

Digit-based shortcut rules for common divisors

  • 2: the last digit is 0, 2, 4, 6, or 8.
  • 3: the sum of the digits is divisible by 3.
  • 4: the last two digits form a number divisible by 4.
  • 5: the last digit is 0 or 5.
  • 6: the number is divisible by both 2 and 3.
  • 7: double the last digit, subtract it from the remaining leading digits, and repeat — if the final result is divisible by 7 (including 0), so is the original number.
  • 8: the last three digits form a number divisible by 8.
  • 9: the sum of the digits is divisible by 9.
  • 10: the last digit is 0.
  • 11: the alternating sum of the digits (from the right) is divisible by 11.
  • 12: the number is divisible by both 3 and 4.
  • 13: multiply the last digit by 4, add it to the remaining leading digits, and repeat — if the final result is divisible by 13, so is the original number.

Common mistakes

  • Treating 6 as a single-digit shortcut: there is no direct digit trick for 6 — you must confirm the number passes both the rule for 2 and the rule for 3.
  • Reusing the digit-sum rule for the wrong divisor: the digit-sum trick only works for 3 and 9 — it does not tell you anything about divisibility by 2, 6, 7, or 11.
  • Confusing quotient and remainder: the quotient is how many whole times D fits into N; the remainder is what is left over. Only a remainder of 0 means "divisible."

Real-world applications

  • Splitting a bill, batch of items, or group of people evenly checks divisibility by the group size.
  • Simplifying fractions relies on finding common divisors of the numerator and denominator.
  • Trial division — testing a number for divisibility by small primes — is the basic method for checking whether a number is prime.
  • Calendars and scheduling use divisibility rules for 4, 100, and 400 to determine leap years.

Frequently Asked Questions

What does it mean for a number to be divisible by another?
A number N is divisible by D when dividing N by D leaves a remainder of 0 — that is, N = D × q for some integer q. For example, 84 is divisible by 3 because 84 = 3 × 28 with a remainder of 0.
How can I tell if a number is divisible by 3 or 9 without dividing?
Add up the digits of the number. If that digit sum is divisible by 3, the original number is divisible by 3; if the digit sum is divisible by 9, the original number is divisible by 9. For 84, the digit sum is 8+4=12, and 12 is divisible by 3, so 84 is divisible by 3.
How does the divisibility rule for 11 work?
Starting from the rightmost digit, alternately add and subtract the digits. If the result (including 0) is divisible by 11, so is the original number. For 121: 1 − 2 + 1 = 0, and 0 is divisible by 11, so 121 is divisible by 11.
Does a divisibility test work for negative numbers?
Yes. Divisibility depends only on whether the remainder is zero, not on the sign of the number. For example, −84 is divisible by 3 for the same reason 84 is, since −84 = 3 × (−28) with a remainder of 0.

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