Remainder Calculator

The remainder calculator finds the quotient and remainder when one integer is divided by another. Integer division asks how many times the divisor fits into the dividend as a whole number, and whatever is left over is the remainder. The relationship is captured in the equation: dividend equals divisor multiplied by quotient plus remainder. For example, when you divide 23 by 4, the divisor 4 fits into 23 exactly five times, leaving a remainder of 3, so the equation reads 23 = 4 x 5 + 3. The remainder is always a non-negative integer strictly less than the divisor when both numbers are positive. You enter the dividend (the number being divided) and the divisor (the number you are dividing by). The calculator returns the quotient, the remainder, the full division equation, and the exact decimal result of the division so you can see how the remainder relates to the fractional part. The tool supports negative integers as well, using truncated division where the quotient is rounded towards zero. Remainders and modulo arithmetic underlie many areas of mathematics and computing: determining whether a number is even or odd (remainder when divided by 2 equals zero or one), checking divisibility, implementing cyclic patterns such as days of the week, and building hash functions. The calculator suits students learning long division, developers debugging modulo logic, and anyone who needs a quick check of a division result.

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3
remainder
Quotient5
Division equation23 = 4 × 5 + 3
Decimal result5.75

How it works

The quotient is the integer part of the division, obtained by truncating (rounding towards zero) the result of dividend divided by divisor. The remainder is then calculated as: remainder = dividend minus (divisor times quotient). This matches the truncated-division convention used by most calculators and programming languages. The full equation is written in the standard form: dividend = divisor x quotient + remainder. The decimal result is the unrounded value of dividend divided by divisor, which shows you exactly where the remainder comes from as the fractional portion.

Worked example

Using the defaults: dividend = 23, divisor = 4. The quotient is the integer part of 23 / 4 = 5.75, which truncates to 5. The remainder is 23 minus (4 x 5) = 23 minus 20 = 3. The full equation is 23 = 4 x 5 + 3. The decimal result is 5.75, confirming the 0.75 fractional part equals 3/4.

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Frequently asked questions

What is a remainder in division?

The remainder is the amount left over after dividing one integer by another as evenly as possible. When you divide 23 by 4, the divisor 4 goes into 23 five times (the quotient) with 3 left over (the remainder), giving the equation 23 = 4 times 5 plus 3. The remainder is always a non-negative integer less than the divisor.

What is the difference between remainder and modulo?

For positive integers, remainder and modulo give the same result. They differ when one of the numbers is negative, depending on the programming language or convention used. The truncated remainder follows the sign of the dividend, while the modulo result follows the sign of the divisor. This calculator uses truncated integer division, which matches the behaviour of most calculators and the JavaScript % operator.

How do you find the remainder without a calculator?

Divide the dividend by the divisor using long division, keeping only the whole-number (integer) part of the quotient. Multiply the quotient by the divisor and subtract from the dividend. What is left is the remainder. For example, 23 divided by 4 is 5 remainder 3 because 4 times 5 is 20, and 23 minus 20 is 3.