How to use the Long Division Calculator
- Enter the dividend (the number being divided).
- Enter the divisor.
- Read the quotient, remainder and decimal result.
- Follow the numbered steps to see how each digit of the quotient was found.
How the calculation works
Long division processes the dividend one digit at a time from the left. At each step you carry the previous remainder, multiply it by ten, add the next digit, and ask how many times the divisor fits. That count becomes the next quotient digit and what is left becomes the new remainder. The algorithm is exactly the same one taught in primary school, formalised.
The remainder is always smaller than the divisor — if it were not, the quotient digit would have been larger. That invariant is what guarantees the process terminates for integer division, and it is the basis of modular arithmetic, where the remainder is the object of interest rather than a leftover.
Continuing past the decimal point appends zeros rather than digits. If a remainder ever repeats, the decimal repeats from that point onward, which is why 1 ÷ 7 produces the six-digit cycle 142857 forever: only six distinct non-zero remainders are possible with divisor 7.
dividend = divisor × quotient + remainder, with 0 ≤ remainder < divisorSource: Euclidean division theorem; standard long division algorithm.
Worked example
Sharing 1,234 pencils equally across 7 classrooms.
- 1 ÷ 7 = 0 remainder 1.
- 12 ÷ 7 = 1 remainder 5.
- 53 ÷ 7 = 7 remainder 4.
- 44 ÷ 7 = 6 remainder 2.
176 pencils per classroom with 2 left over (1,234 ÷ 7 = 176.286).
Frequently asked questions
Why is the remainder always less than the divisor?+
Because if it were larger, the divisor would fit at least once more and the quotient digit would increase.
How do I get a decimal instead of a remainder?+
Keep dividing, appending zeros after the decimal point. The decimal row shows the result directly.
What makes a decimal repeat?+
A repeated remainder. Only finitely many remainders are possible, so any non-terminating decimal must eventually cycle.
Can the divisor be zero?+
No. Division by zero is undefined.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.