Remainder Calculator

Dividing a whole number a by a whole number b gives a whole-number quotient q and a remainder r that is smaller than the divisor:

a = b × q + r, with 0 ≤ r < b

For any positive a and b there is exactly one such pair, which is why the result is well defined. 157 ÷ 13 is 12 remainder 1, because 13 × 12 = 156 and one is left over. The calculator gives the quotient, the remainder and a check line, then continues the division past the decimal point, writes the result as a mixed number (12 1/13) and as a decimal, and lays out the full long division the way it is written by hand.

Long division repeats four moves for each digit of the dividend: divide (how many times does the divisor fit?), multiply, subtract, and bring down the next digit. Here is 9,876 ÷ 24 as the calculator works it:

1. Start with 98. 24 fits 4 times; 4 × 24 = 96; 98 − 96 = 2.
2. Bring down the next digit to make 27. 24 fits 1 time; 1 × 24 = 24; 27 − 24 = 3.
3. Bring down the next digit to make 36. 24 fits 1 time; 1 × 24 = 24; 36 − 24 = 12.

The quotient digits read 411, and the last subtraction leaves the remainder 12. Check: 24 × 411 + 12 = 9,876. The first digit, 9, is smaller than 24, so the work starts with 98; a leading zero in the quotient is never written. Every subtraction must leave less than the divisor; if it does not, the quotient digit was too small.

To turn the remainder into decimals, put a decimal point after the dividend and keep bringing down zeros. Because each remainder is smaller than the divisor, there are only so many possible remainders; as soon as one repeats, the digits repeat too. A fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5. The calculator finds the repeating block, shown in brackets:

The repeating block can never be longer than the divisor minus one, which is why 1 ÷ 7 repeats after six digits. Choose how many decimal places to include in the written layout with the long division setting.

Long division needs a whole-number divisor, so a decimal divisor is cleared first by moving the decimal point in both numbers the same number of places. 7.5 ÷ 0.4 becomes 75 ÷ 4, which gives the same quotient, 18.75. The whole-number quotient is 18 and the remainder is 0.3, in the original units: 7.5 = 18 × 0.4 + 0.3.

With negative numbers the division is worked on the sizes and the sign applied afterward. The calculator's main answer rounds the quotient toward zero, as C, Java and JavaScript do: −17 ÷ 5 = −3 R −2. Python and spreadsheets round down instead, giving −4 R 3, and that version is shown too. The modulo calculator compares all the sign conventions in detail.

A remainder of zero means the divisor goes in exactly. The divisibility list divides your dividend by each number you enter. For 360 and the divisors 2 to 12, everything divides evenly except 7 and 11, which is part of why 360 was a convenient number of degrees for a full turn.

Quick mental checks give the same answers for small divisors: a number is divisible by 3 or 9 when its digit sum is, by 4 when its last two digits are, by 5 when it ends in 0 or 5, and by 11 when the alternating sum of its digits is a multiple of 11.