Remainder Calculator - Quotient, Remainder, and Steps

A remainder calculator finds the leftover after one integer is divided by another. It is the tool to reach for when a long division answer leaves a value that does not split cleanly, when you need the integer quotient together with the leftover, or when you want to verify a result with the dividend = quotient times divisor plus remainder identity. This remainder calculator handles the small everyday case (17 divided into 5 equal groups) and the larger cases that show up in modular arithmetic and number theory.

• • School division problems: Get the integer quotient and leftover for any dividend and divisor pair used in grade-school long division.
• • Long division verification: Check a long division answer by multiplying the quotient by the divisor and adding the leftover to confirm the original dividend.
• • Modular arithmetic setup: Compute the remainder of a division before chaining it into a modular inverse or modular exponentiation workflow.
• • Counting and grouping tasks: Decide how many complete groups fit into a quantity and how many items are left over for seating, packing, or scheduling.

The result panel returns the integer quotient, the leftover remainder, the decimal remainder (the fraction of the divisor that is left), and a verification line that multiplies the quotient by the divisor and adds the remainder. That verification line should always equal the dividend you typed in.

If the dividend is negative, the answer depends on the convention you pick. Floor convention follows the mathematical definition used in number theory. Truncated convention matches what most programming languages do, including the remainder operator in C, Java, and JavaScript.

For the same leftover in a modular arithmetic context, including negative-number handling and modular inverses, the Modulo Calculator extends this remainder result into the full cycle.

The math is the same as the long division algorithm: divide, take the whole-number part, multiply it back by the divisor, and subtract. The remainder is whatever is left after the subtraction.

• Dividend: The number being divided. It can be any integer, including negative values, and the calculator will still return an exact remainder.
• Divisor: The number dividing the dividend. The remainder is always smaller than the absolute value of the divisor in floor convention. Zero is not allowed.
• Quotient: The whole-number part of the division. With the floor convention, quotient is rounded toward negative infinity; with the truncated convention, it is rounded toward zero.
• Remainder: The leftover after the quotient is taken out. The verification identity dividend = quotient * divisor + remainder is always true.

Long division arrives at the same answer by repeated subtraction. Each digit you bring down is one more step of taking out groups of the divisor.

With the floor convention, negative dividends stay clean: -17 divided by 5 gives quotient -4 and remainder 3, because -4 * 5 + 3 = -17. The remainder is non-negative whenever the divisor is positive. According to Wolfram MathWorld, the remainder is the unique integer r such that a = qd + r for integer dividend a, integer divisor d, and integer quotient q.

According to Wolfram MathWorld, the remainder is the unique integer r such that a = qd + r for integer dividend a, integer divisor d, and integer quotient q, with 0 <= r < |d| when the floor convention is used.

When you want to see the digit-by-digit long division that produces the same quotient and remainder, the Long Division Calculator walks through the same algorithm step by step.

Four ideas sit behind every division with a leftover. Keep them separate and the remainder calculator answer is easy to interpret.

The decimal remainder is a useful extra: it is remainder divided by divisor, so it tells you the fraction of the divisor that did not fit. For 17 / 5, the decimal remainder of 0.4 means 40 percent of the divisor is still left over.

If you are working with primes, the remainder is also a divisibility test: a number is divisible by n only when the remainder of dividend divided by n is zero. The same idea shows up in factor checks and in coprime tests.

Because a number is divisible by n only when the remainder of dividend divided by n is zero, the Prime Number Checker is a natural next stop after checking a leftover.

Type the dividend and divisor, choose a convention for negatives, and read the four outputs that appear in the result panel of this remainder calculator.

1. 1 Enter the dividend: Type the number being divided in the first field. The calculator accepts negative integers, decimals, and zero.
2. 2 Enter the divisor: Type the number that will divide the dividend. The divisor must be a non-zero number; zero returns a clear error.
3. 3 Pick a remainder convention: Use floor for math and number theory. Use truncated when matching a programming language like C, Java, or JavaScript.
4. 4 Read the outputs: The result panel shows the integer quotient, the leftover remainder, the decimal remainder, and a verification line of q * d + r that should match the dividend.
5. 5 Verify the answer: If q * d + r on the result panel does not equal the dividend you typed in, switch the convention or re-check the inputs.

Suppose 100 students are seated in tables of 13. The integer quotient of 100 / 13 is 7, so 7 full tables are used. The remainder is 100 - 7 * 13 = 12, so 12 students are seated at an eighth table. The decimal remainder is 12 / 13.

When the same quotient idea appears with fraction inputs instead of integers, the Divide Fractions Calculator handles the keep-change-flip rule while this page keeps the integer leftover.

The remainder is small math, but it shows up in many workflows where a clean answer changes the next decision.

• • Check long division answers: Use the verification line to confirm a hand-computed long division result in one click.
• • Pick the right convention: Switch between floor and truncated remainders to match what a textbook, a paper, or a programming language expects.
• • Plan groups and leftovers: See exactly how many full groups fit into a quantity and how many items are left over for seating, packing, and scheduling.
• • Bridge to modulo and primes: Use the leftover as the starting point for a modular inverse, a modular exponentiation, or a divisibility test on the same dividend and divisor.
• • Avoid decimal rounding errors: Compare the exact integer remainder with the decimal remainder so you can tell when a rounded decimal is hiding a clean division.

For day-to-day planning, the integer quotient and the leftover are usually the two numbers you need. The quotient tells you the number of complete groups. The leftover tells you what still needs a partial group.

For number theory and programming, the convention matters. Pick floor for math papers, the Euclidean algorithm, and the existing modulo-calculator. Pick truncated for code reviews and language comparisons.

For the same quotient-and-remainder idea applied to polynomial division by a linear term, the Synthetic Division Calculator uses a different layout to produce the polynomial quotient and remainder.

The remainder can shift with the sign of the dividend, the sign of the divisor, and the convention you pick.

• • The calculator only handles integer-style divisions. Polynomial, matrix, and fraction division each follow their own rules and need a dedicated tool.
• • The remainder is not the same as a modular inverse. The leftover tells you what is left after the largest possible whole number of groups; an inverse finds a multiplier that returns 1, which only exists when the dividend and divisor are coprime.
• • The decimal remainder is rounded to 6 decimal places for readability. The integer quotient, integer remainder, and verification identity are exact, but the decimal part is a presentation of the same exact fraction.

The convention choice is a presentation issue more than a math issue, but it still matters when answers get compared. According to Britannica, the division algorithm expresses the dividend as a product of the divisor and the quotient plus a remainder that is smaller in absolute value than the divisor, which is the floor convention.

When the divisor and dividend are coprime, the leftover is the starting point for a modular inverse. The Euclidean algorithm uses repeated remainders to find a multiplier that returns 1.

According to Britannica, the division algorithm expresses the dividend as a product of the divisor and the quotient plus a remainder that is smaller in absolute value than the divisor.

According to Khan Academy, the remainder is what is left over after dividing the dividend into the largest possible whole groups of the divisor, and you can verify it with quotient times divisor plus remainder equals the dividend.

When the dividend and divisor are polynomials instead of integers and the remainder is still a polynomial, the Polynomial Division Calculator runs the same kind of divide-and-subtract algorithm symbolically.