Remainder Calculator

This remainder calculator gives the quotient and the remainder for any division, and shows what happens when either number is negative. You get the truncated remainder that C, Java and JavaScript return, the floored modulo Python and Ruby return, and the Euclidean remainder.

How to use the remainder calculator

1. Type the dividend — the number being divided — into the first box.
2. Type the divisor into the second box. Either value can be negative.
3. Read the remainder at the top, with the full statement underneath in the form 17 ÷ 5 = 3 remainder 2.
4. The panel below lists the quotient, the floored modulo, the Euclidean remainder and the exact decimal quotient, so you can see all three conventions side by side.

Results update as you type. If you want the decimal answer worked out in full instead of the leftover, the division calculator takes the same division through long division and into the decimal places.

What a remainder actually is

That statement is called the division algorithm, and the uniqueness is the point. For 17 and 5 the only whole-number pair that works is q = 3 and r = 2, because 5 × 3 + 2 = 17 and 2 is smaller than 5.

The remainder is what is left after you take out as many whole divisors as will fit. It is always smaller than the divisor. If you ever produce a remainder as big as the divisor, another whole one fits and the quotient was too low.

There is also a quick way to get a remainder from any plain calculator. Divide, throw away everything after the point, multiply back by the divisor and subtract: 100 ÷ 12 = 8.333…, so 12 × 8 = 96 and 100 − 96 = 4.

Remainder and modulo: the same until something goes negative

For positive numbers, “remainder” and “modulo” mean the same thing and every calculator, language and textbook agrees. Introduce a negative and they split into three different answers, all of them defensible.

• Truncated remainder. Cut the quotient toward zero, then subtract. The result takes the sign of the dividend. This is what C, C++, Java, C#, Go, Rust and JavaScript’s % return.
• Floored modulo. Round the quotient down toward negative infinity. The result takes the sign of the divisor. This is Python’s %, Ruby’s %, and what mathematicians usually mean by “mod n”.
• Euclidean remainder. Always non-negative, whatever the signs. This is the one that satisfies 0 ≤ r < |b| from the division algorithm above.

Take the second row. JavaScript evaluates -17 % 5 as −2, because it truncates −3.4 to −3 and 5 × −3 = −15 leaves −2. Python evaluates -17 % 5 as 3, because it floors −3.4 to −4 and 5 × −4 = −20 leaves 3. Both satisfy a = b × q + r. They just chose different qs.

What remainders are used for

Remainders are far more than the leftover bit of a school division. Most of the time you want the remainder and do not care about the quotient at all.

• Odd or even. n mod 2 is 0 for even numbers and 1 for odd. The same idea colors alternate rows of a table or splits people into two teams.
• Every nth item. i mod 3 = 0 picks out every third item in a list, which is how repeating patterns and page breaks get built.
• Leap years. A year is a leap year if it is divisible by 4, unless it is divisible by 100, unless it is also divisible by 400. 2024 mod 4 = 0, so it is a leap year. 1900 mod 100 = 0 and 1900 mod 400 = 300, so it is not. 2000 mod 400 = 0, so it is.
• Clocks and wrap-around. Seven hours after 21:00 is (21 + 7) mod 24 = 4, so 04:00. One hundred days after a Tuesday lands on 100 mod 7 = 2 days later, a Thursday.
• Splitting into groups. 100 items packed 12 to a box is 8 full boxes with a remainder of 4, so you need 9 boxes and the last one is part-filled.
• Check digits. Bank codes, barcodes and ISBNs finish with a digit chosen so that a weighted sum of all the digits leaves remainder 0 when divided by 10, 11 or 97. Mistype one digit and that remainder is no longer 0, so the number is rejected.

Hashing works the same way: mod table_size picks a slot. For very large values, exact remainders need exact arithmetic, which the big number calculator gives you.

Divisibility rules: predicting a zero remainder

You can often tell that the remainder will be 0 without dividing at all. These rules are exact, not rules of thumb.

The 3 and 9 tests work because 10 leaves remainder 1 when divided by 3 or by 9, so every power of ten does too, and a number ends up with the same remainder as the sum of its digits. The 11 test works because 10 leaves remainder −1 when divided by 11, so the place values alternate in sign.

The digit-sum tests give you more than a yes or no. The remainder of the digit sum is the remainder of the whole number: 2,573 has digit sum 17, and 17 mod 3 = 2, so 2,573 mod 3 = 2 as well.

Common mistakes

• Reading the decimal part as the remainder. 17 ÷ 5 = 3.4 does not mean remainder 4. The 0.4 is a fraction of the divisor, so the remainder is 0.4 × 5 = 2.
• Assuming every language agrees on negatives. -17 % 5 is −2 in JavaScript and 3 in Python. Porting code between the two without checking will move an index or a date by a whole cycle.
• Producing a remainder as big as the divisor. If you get remainder 5 when dividing by 5, one more divisor fits. The quotient was one too small.
• Rounding the quotient instead of truncating it. The quotient is the whole part, rounded toward zero, not to the nearest whole number.
• Dropping the remainder when it needs a whole extra unit. 100 people at 12 per bus needs 9 buses, not 8.
• Taking a remainder mod 0. There is no such thing, for the same reason division by zero is undefined.
• Forgetting the check. Multiply the divisor by the quotient and add the remainder with the multiplication calculator. If you do not get the dividend back, one of the two is wrong.

Frequently asked questions

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