Modulo Calculator

1. Enter the dividend a, the number being divided.
2. Enter the divisor b, which cannot be zero.
3. Read a mod b, the remainder after division.
4. See the quotient and the truncated remainder for comparison.

Here is what each result means:

The modulo operation gives the remainder after one number is divided by another. If you divide a by b, the modulo is what is left over. So 17 mod 5 = 2, because 5 goes into 17 three times with 2 remaining. When the remainder is zero, a is exactly divisible by b.

Modulo is everywhere in maths and computing. It tells you whether a number is even or odd, wraps a clock around after 12 hours, cycles through colors or list positions, and drives hashing and cryptography. It turns a straight number line into a repeating cycle of length b.

It divides a by b, takes the whole-number quotient, and returns what is left.

1. Find how many whole times b fits into a.
2. Multiply that quotient by b and subtract from a.
3. The leftover is the remainder; for a positive divisor the modulo is never negative.

Modulo underpins the greatest common divisor; the GCD calculator uses repeated remainders to find it.

Find 17 mod 5 and check whether 100 is even.

Calculation: 5 fits into 17 three times, using 15, so 17 mod 5 = 2. For 100, 100 mod 2 = 0, and a remainder of 0 means 100 is even. Any number with a remainder of 1 when divided by 2 is odd.

The two agree for positive numbers but can differ when the dividend is negative.

Most programming languages return the truncated remainder, which keeps the sign of the dividend. The mathematical modulo keeps the sign of the divisor, so it is never negative for a positive b.

The repeating nature of modulo shows up in many places.

Anywhere a count needs to wrap around and repeat, modulo is the natural tool.

The Size of the Divisor

The remainder is always smaller than the divisor, so a larger b allows a wider range of remainders.

The Sign of the Numbers

A negative dividend or divisor is where the truncated remainder and the mathematical modulo can differ.

Exact Division

When b divides a with nothing left over, both the remainder and the modulo are zero.

Programming and Logic

Check divisibility, wrap indexes, or work out cycles.

Number Theory Homework

Find remainders and explore divisibility rules.

Everyday Cycles

Work out clock arithmetic, repeating patterns or check digits.

1. Dividing by Zero

The divisor cannot be zero, since division by zero is undefined.

2. Expecting a Negative Modulo

For a positive divisor, the mathematical modulo is never negative, even when the dividend is negative.

3. Confusing Remainder and Quotient

The quotient is how many times b fits; the remainder is what is left over.

4. Assuming All Languages Agree

Programming languages differ on negatives. Check whether yours truncates or floors.

5. Thinking Modulo Needs Whole Numbers

Modulo also works on decimals, giving a decimal remainder smaller than the divisor.

The remainder is exact for whole numbers; decimals may carry tiny floating-point effects.

It gives the remainder after dividing one number by another. So 17 mod 5 is 2, because 5 fits into 17 three times with 2 left over. A remainder of zero means the division is exact.

Divide a by b, take the whole-number quotient, multiply it back and subtract from a. Whatever is left is the remainder. This tool does it and shows the quotient too.

They agree for positive numbers. For negatives they can differ: the truncated remainder keeps the sign of the dividend, while the mathematical modulo keeps the sign of the divisor and is never negative for a positive b.

For a positive divisor, the mathematical modulo is defined to fall between zero and the divisor. So even a negative dividend gives a non-negative result, which is why -17 mod 5 is 3, not -2.

Take the number mod 2. If the result is 0 it is even, and if it is 1 it is odd. The same idea checks divisibility by any number.

Yes. The modulo of decimals gives a decimal remainder smaller than the divisor. For example, 5.5 mod 2 is 1.5. It is most common with whole numbers, though.

It checks divisibility, wraps list indexes so they stay in range, cycles through repeating patterns, and powers hashing and check digits. It is one of the most-used operators in code.

Some truncate the quotient toward zero and some floor it, which changes the sign of the remainder for negative inputs. This tool shows both so there is no confusion.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Modulo (Wikipedia).
• The modulo operation (Maths Is Fun).
• Modular arithmetic (Wikipedia).