The Remainder Calculator finds the quotient and the remainder when one whole number (the dividend) is divided by another (the divisor). Enter both numbers and click Calculate.
The remainder calculator works out what is left over when you divide one number by another. Enter the dividend (the number being divided) and the divisor (the number you are dividing by), and it returns the quotient — how many whole times the divisor goes in — and the remainder, the amount that does not divide evenly. It also shows the complete division equation and the decimal result, so you can see exactly how the pieces fit together.
Remainders are one of the first ideas you meet in arithmetic, back when division is taught as "how many times does this go in, and what's left over." But they never stop being useful: remainders power everything from telling time and scheduling to computer science, cryptography, and number theory. This calculator handles the arithmetic instantly for numbers of any size.
When you divide one integer by another, the result is not always a whole number. The remainder is the integer amount left over after you have taken out as many whole copies of the divisor as possible. For example, if you divide 13 by 4, the number 4 goes in 3 whole times (3 × 4 = 12), and there is 1 left over. So the quotient is 3 and the remainder is 1.
A remainder is always smaller than the divisor — if it were equal to or larger, the divisor would go in at least one more time. When the remainder is zero, the division is exact and we say the dividend is divisible by the divisor. The remainder is what distinguishes ordinary integer division from the exact division that produces fractions or decimals.
The calculator uses the standard definition of integer division. Given a dividend a and a divisor n, it finds the quotient q and the remainder r so that:
a = n × q + r
The steps are simple: divide the dividend by the divisor, round the result down to the nearest whole number to get the quotient, multiply that quotient by the divisor, and subtract from the dividend to get the remainder. In formula terms, q = ⌊a ÷ n⌋ and r = a − n × q. For 487 divided by 32: 487 ÷ 32 = 15.22, which rounds down to a quotient of 15; then 487 − 32 × 15 = 487 − 480 = 7, so the remainder is 7. The full equation reads 487 = 32 × 15 + 7.
The same relationship is often written as a / n = q + r/n, which just expresses the division as a whole-number part plus a fractional part. For it to make full sense, the dividend and divisor should be integers — the concept of a "remainder" is defined for whole numbers — though the arithmetic itself will still produce a consistent answer for other inputs.
It helps to keep the four terms straight, because they come up constantly:
You will often see integer division written with an "R" for remainder, as in 13 ÷ 4 = 3 R 1. That shorthand is exactly what this calculator produces, along with the fuller equation form.
The remainder is the heart of modular arithmetic, sometimes called "clock arithmetic." When mathematicians write "a mod n," they mean the remainder when a is divided by n. This idea is far more powerful than it first appears. A 12-hour clock is modular arithmetic in action: 5 hours after 9 o'clock is not 14 o'clock but 2 o'clock, because 14 mod 12 = 2. Days of the week work the same way with mod 7, and so do many repeating cycles.
The "mod" operation underpins a huge amount of modern technology. It is central to cryptography — the encryption that secures online banking and messaging relies on modular exponentiation of very large numbers. It is used in hashing, which distributes data evenly across storage buckets, and in checksums and error detection, where remainders verify that data has not been corrupted. Even the check digit on a barcode or credit card number is computed with a remainder. Understanding remainders is the first step toward all of these applications.
There are three ways to express the result of a division that does not come out even, and each is useful in different situations. Take 13 ÷ 4:
They all describe the same division; the remainder form simply keeps everything as whole numbers. The relationship between them is direct: the decimal's fractional part (0.25) equals the remainder divided by the divisor (1 ÷ 4), and the fraction (¼) is that same remainder over the divisor. This calculator gives you the whole-number quotient and remainder, and also shows the decimal so you can compare.
Division with negative numbers introduces a subtlety, because there is more than one convention for what the remainder should be. This calculator rounds the quotient down (toward negative infinity), which is the convention used in mathematics and in many programming languages such as Python. Under this rule, the remainder always takes the same sign as the divisor. For example, −13 divided by 4 gives a quotient of −4 (since −3.25 rounds down to −4) and a remainder of 3, because −13 = 4 × (−4) + 3. Some other systems round the quotient toward zero instead, which would give a quotient of −3 and a remainder of −1. Both satisfy the equation a = n × q + r; they simply differ in how they handle the sign. For ordinary positive whole numbers, none of this ambiguity arises, and the answer is always straightforward.
Before calculators, the remainder was found by long division, the step-by-step written method taught in school. You work through the dividend digit by digit, at each stage seeing how many times the divisor fits, writing that above the line, subtracting, and bringing down the next digit. When you run out of digits, whatever is left at the bottom is the remainder. Long division is still valuable for understanding why the quotient and remainder are what they are, and for handling very large numbers by hand. This calculator automates the process and gives you the same quotient and remainder instantly, but the underlying logic is exactly the long division you may remember.
The rule behind this calculator has a formal name in mathematics: the division algorithm. It states that for any integer dividend a and any positive integer divisor n, there exist unique integers q (the quotient) and r (the remainder) such that a = n × q + r, with the remainder constrained to 0 ≤ r < n. The word "unique" is the important part: there is exactly one pair of quotient and remainder that satisfies both the equation and the size restriction on r. This guarantee is what makes remainders so dependable as a mathematical tool — every division of whole numbers has one and only one correct answer in quotient-and-remainder form, and it is the foundation on which number theory, the Euclidean algorithm for finding greatest common factors, and modular arithmetic are all built.
Remainders give a precise way to talk about divisibility. Saying "6 divides 24" is the same as saying "the remainder of 24 ÷ 6 is 0." This connects directly to the divisibility rules you may have learned: a number is divisible by 2 when its remainder mod 2 is 0 (it is even), by 5 when it ends in 0 or 5, by 3 when the sum of its digits leaves no remainder mod 3, and so on. Each rule is really a shortcut for checking whether a certain remainder is zero. Prime numbers can be described the same way — a prime greater than 1 is a number that leaves a non-zero remainder when divided by every integer between 2 and itself. Whenever you need to know if one number goes evenly into another, the remainder is the definitive test, and a result of zero from this calculator is the confirmation.
Divide the dividend by the divisor and round down to get the quotient. Multiply the quotient by the divisor and subtract that from the dividend — what is left is the remainder. In symbols, r = a − n × ⌊a ÷ n⌋. The calculator above does this for you.
The quotient is 15 and the remainder is 7, because 32 × 15 = 480 and 487 − 480 = 7. Written out, 487 = 32 × 15 + 7.
A remainder of zero means the division is exact — the dividend is perfectly divisible by the divisor with nothing left over. For example, 24 ÷ 6 = 4 remainder 0, so 24 is a multiple of 6.
No. The remainder is always smaller than the divisor. If it were equal to or greater than the divisor, the divisor would fit into the dividend at least one more time, which would increase the quotient and reduce the remainder.
They are closely related — "a mod n" is the remainder when a is divided by n. For positive numbers they are identical. They can differ for negative numbers, where different rounding conventions give differently signed remainders. This calculator uses the round-down convention, so the remainder matches the sign of the divisor.
The concept of a remainder is defined for whole numbers, so integer inputs are recommended. The calculator will still apply the formula consistently to other inputs, but the "remainder" only has its usual meaning when both the dividend and divisor are integers.
This Remainder Calculator is provided for educational and general informational purposes. It computes the quotient and remainder of integer division using the standard round-down convention. Results are exact for whole-number inputs.