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Math Calculator

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Euclidean division:
a = bq + r, with 0 ≤ r < |b| (b ≠ 0)

Remainder is computed using |b| so it is always non-negative.

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How to Use the Remainder Calculator

A remainder calculator helps you compute the remainder when one integer is divided by another. The remainder is what “is left over” after dividing as many whole times as possible. For example, 23 divided by 5 gives quotient 4 and remainder 3, because 23 = 5×4 + 3.

This calculator uses Euclidean division, which always returns a non-negative remainder r such that 0 ≤ r < |b| (as long as the divisor b is not zero). That is especially helpful when working with negative numbers, because different programming languages can define the sign of % differently.

Enter the dividend a and divisor b as integers (you can use negative values too). Click Calculate to see both the quotient q and remainder r, along with the identity a = |b|·q + r. The remainder shown is always in the standard Euclidean range.

Remainders are used in modular arithmetic, number theory, cyclic patterns (like clocks), hashing, and many programming tasks. If you need a remainder for polynomial division by (x − a), use the Polynomial Remainder Calculator instead.

Example

Dividend a = 23, divisor b = 5 → quotient q = 4, remainder r = 3, because 23 = 5×4 + 3. If a = -23 and b = 5, Euclidean remainder is r = 2, since -23 = 5×(-5) + 2.

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Frequently Asked Questions

When you divide an integer a by a non-zero integer b, you can write a = bq + r. The remainder r is the part left over after taking q whole multiples of b. In Euclidean division, r is always non-negative and smaller than |b|.
People often use the terms interchangeably, but in some contexts “modulo” specifically means the Euclidean remainder (0 ≤ r < |b|). Some programming languages define % in a way that can produce negative results when a is negative. This calculator shows the Euclidean (non-negative) remainder.
No. Division by zero is undefined, so the calculator will show an error if the divisor is 0.
Yes. You can enter negative integers for the dividend and/or divisor. The calculator still returns a non-negative remainder using |b| so the remainder stays in the Euclidean range.
Because the calculator uses Euclidean division, where the remainder is defined to satisfy 0 ≤ r < |b|. This avoids ambiguity with negative inputs and matches common math definitions used in modular arithmetic.