Remainder Calculator — Integer Division with Steps
Divide any two whole numbers to get the quotient and remainder — with the full a = b × q + r breakdown shown step by step and illustrated on a number line.
47 = 7 × 6 + 5
Set up the division: 47 ÷ 7
Find the quotient (ignore the fractional part)
Compute the remainder using a = b × q + r
Verify
- 1
Quotient (integer part of a ÷ b)
int(47 ÷ 7) = 6 - 2
Product b × q
7 × 6 = 42 - 3
Remainder r = a − b × q
47 − 42 = 5Check: b × q + r = a, so the identity a = b × q + r holds.
How does this calculator work?
For integers a ÷ b: quotient q = trunc(a / b), remainder r = a − b×q, and the check is a = b×q + r. For 47 ÷ 7: q = 6, r = 5, check: 7×6 + 5 = 47. The remainder always satisfies |r| < |b|. The modulo is the always-non-negative version of the remainder.
Formula
How this is calculated
Integer division splits a dividend a by a divisor b into a quotient q (how many times b fits into a) and a remainder r (what is left over). The fundamental relationship is always a = b × q + r. This calculator uses truncated division — q is the integer part of a/b with the fractional part discarded — which matches the behaviour of the % operator in most programming languages (C, Java, JavaScript, Python 3 for positive inputs).
For positive inputs the remainder r satisfies 0 ≤ r < b, and the modulo and remainder are identical. For negative dividends or divisors the two can differ: the modulo is always non-negative (it wraps to the positive side), while the remainder keeps the sign of the dividend. This distinction matters in programming: Python's % operator uses floored (modulo) division; C, Java and JavaScript use truncated (remainder) division.
The number line visualization shows where a and b × q lie — the gap between them is exactly the remainder r, making the a = b × q + r identity geometrically obvious.
Frequently asked questions
For positive numbers they are identical. They differ when the dividend is negative: with 47 ÷ −7, the truncated remainder is 5 (keeps the sign of the dividend), while the modulo is −2 (keeps the sign of the divisor). Most calculators and spreadsheet MOD() functions use the modulo (floored) convention.
This is the Division Algorithm: any integer a can be uniquely written as b multiplied by an integer quotient q plus a remainder r, where the remainder is strictly smaller in absolute value than the divisor. For example, 47 = 7 × 6 + 5.
Remainders appear throughout mathematics and computing: checking divisibility (a is divisible by b if r = 0), cyclic patterns (clock arithmetic, day-of-week calculations), hashing and modular arithmetic in cryptography, and as the core operation of the Euclidean GCD algorithm.
Also known as
TG we-Calculate Editorial Team. (2026). Remainder Calculator — Integer Division with Steps [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/remainder-calculator
TG we-Calculate Editorial Team. "Remainder Calculator — Integer Division with Steps." TG we-Calculate. 2026. https://we-calculate.com/calculator/remainder-calculator.
TG we-Calculate Editorial Team, "Remainder Calculator — Integer Division with Steps," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/remainder-calculator
@misc{wecalculate_remainder_calculator, title = {Remainder Calculator — Integer Division with Steps}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/remainder-calculator}}, year = {2026}, note = {TG we-Calculate} }
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