Divisibility Test Calculator

What this calculator does

So, you want to know whether one number divides another evenly, with nothing left over. This tool tells you. Enter a number and a divisor, and it says whether the first is divisible by the second, and if so, gives you the result of the division.

Two boxes, one for the number and one for the divisor, and a single button.

How to use it

1. Enter the number you want to test.
2. Enter the divisor to test it against.
3. Press Calculate.

What divisibility means

One number is divisible by another when dividing them leaves no remainder, so the result is a whole number. 20 is divisible by 4, because 20 divided by 4 is exactly 5, with nothing left over. But 20 is not divisible by 3, because 20 divided by 3 is 6 with a remainder of 2. Divisibility is the plain idea of one number fitting into another a whole number of times, and it underlies factors, primes, and much of the rest of number work.

How the tool checks: the remainder

The tool answers the question in the most direct way, by looking at the remainder. It divides your number by the divisor and checks what is left. If the remainder is zero, the division is exact, so the number is divisible, and the tool also shows you the quotient, the whole number you get. If the remainder is anything other than zero, the number is not divisible by that divisor, and the tool says so. This is the definition of divisibility applied literally: divisible means a remainder of zero.

The divisibility rules, as shortcuts

While the tool does a direct division, it is worth knowing the mental shortcuts, the divisibility rules, that let you check common divisors at a glance. A number is divisible by 2 if its last digit is even. By 5 if it ends in 0 or 5, and by 10 if it ends in 0. By 3 if the sum of its digits is divisible by 3, and by 9 if the digit sum is divisible by 9. By 4 if its last two digits form a number divisible by 4. These rules come from how our base-ten number system works, and they turn many divisibility questions into something you can answer in your head, with the tool there to confirm.

A worked example

Enter a number of 20 and a divisor of 4. The tool divides and finds no remainder, so it reports that 20 is divisible by 4, and shows that 20 divided by 4 is 5. Enter 20 and a divisor of 3 instead, and since the division leaves a remainder of 2, the tool reports that 20 is not divisible by 3.

Why the divisor cannot be zero

The tool will not let you use zero as the divisor, and there is a firm reason. Dividing by zero has no meaning in mathematics: there is no number that zero can be multiplied by to give a nonzero result, so the question "is this number divisible by zero?" has no sensible answer. Rather than return something misleading, the tool asks for a divisor that is not zero. Any other whole number is fair game.

Questions people ask

What does divisible mean?

That dividing leaves no remainder, so the result is a whole number. 20 is divisible by 4 because 20 divided by 4 is exactly 5.

How does the tool decide?

It checks the remainder after dividing. A remainder of zero means divisible, and it shows the quotient; anything else means not divisible.

What are the divisibility rules?

Mental shortcuts, like: even last digit means divisible by 2, a digit sum divisible by 3 means divisible by 3, and ending in 0 or 5 means divisible by 5.

Why can't the divisor be zero?

Because dividing by zero has no meaning, so divisibility by zero is undefined. The tool requires a nonzero divisor.

How does this relate to factors?

If a number is divisible by a divisor, that divisor is a factor of it. Divisibility is the test behind finding factors and primes.

References

On divisibility. A number is divisible by another when the division leaves no remainder, and simple digit rules can test common divisors.

1. Eric W. Weisstein, "Divisibility Tests," from MathWorld, a Wolfram resource, on the digit-based rules for divisibility by small numbers.
2. Eric W. Weisstein, "Prime Factorization Algorithms," from MathWorld, a Wolfram resource, on trial division, which is repeated divisibility checking.