Remainder Calculator

A remainder is the amount left after one integer is divided into as many complete groups of another integer as possible. Integer division can be summarized by the division algorithm: the dividend equals the divisor multiplied by the whole-number quotient, plus the remainder. For 17 divided by 5, three complete groups of 5 use 15 units and leave 2, so the quotient is 3 and the remainder is 2. This calculator uses truncating integer division. It divides the dividend by the divisor, removes the fractional part toward zero to obtain the quotient, and subtracts the quotient times the divisor from the dividend. For positive inputs this matches the familiar long-division convention, where the remainder ranges from zero through one less than the divisor. Exact division produces a remainder of zero and confirms that the divisor is a factor of the dividend. Signed values require a stated convention because mathematics and programming systems do not all define negative remainders in the same way. With truncation toward zero, the remainder has the same sign as the dividend or is zero. For example, negative 17 divided by 5 gives quotient negative 3 and remainder negative 2, satisfying negative 17 equals 5 times negative 3 plus negative 2. Euclidean division instead adjusts the quotient to keep the remainder nonnegative. Both conventions preserve the division identity, but this calculator explicitly uses truncation so its results are predictable. Remainders have many practical uses. They reveal whether a number is divisible by another, alternate events in repeating cycles, map positions around a clock, distribute objects into equal groups, and support modular arithmetic. If 53 students are arranged in rows of 8, the quotient says six full rows can be formed and the remainder says five students remain. A remainder of zero means every item fits evenly. Division by zero is undefined because no finite quotient can satisfy the division identity, so the calculator rejects a zero divisor. Inputs must be integers because a whole-number quotient and remainder are concepts from integer division. For ordinary decimal division, use a standard calculator instead. The equation displayed below each result provides a quick check: multiply the divisor by the quotient, add the remainder, and you must recover the original dividend.