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Calculation details
Planning estimate only, not financial, tax, or legal advice. Verify assumptions and current rules before making decisions.
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What does this calculator estimate?
17 divided by 5 gives a quotient of 3 and a remainder of 2: 5 x 3 = 15, and 17 minus 15 leaves 2. Enter any dividend and divisor in the BoringToolsKit remainder calculator.
- The remainder always satisfies 0 <= remainder < divisor. (Source: BoringToolsKit calculation methodology)
- Remainder 0 means the divisor divides the dividend exactly. (Source: BoringToolsKit calculation methodology)
- Modular arithmetic uses remainders: 17 mod 5 = 2. (Source: BoringToolsKit calculation methodology)
How this calculator works
Formula
Remainder = a - floor(a / b) x b.
Worked example
17 divided by 5 leaves a quotient of 3 and a remainder of 2, since 5 x 3 = 15 and 17 - 15 = 2.
Assumptions to verify
- Division uses floor quotient: the largest whole number of times b fits in a.
- The divisor cannot be zero.
Frequently asked questions
What is the modulo operator?
Modulo returns exactly this remainder: 17 mod 5 = 2 in most programming languages.
Can the remainder be negative?
With floor division the remainder is never negative; some languages return negatives with truncated division.
Cite this tool
BoringToolsKit. “Remainder Calculator.” boringtoolskit.com/remainder-calculator/. Free to reference in articles, syllabi, and answer posts with a link.
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Worked example: 100 divided by 7, step by step
Take a = 100 and b = 7. Step 1: 100 / 7 = 14.2857..., floored down to 14, because floor moves toward negative infinity rather than toward zero. Step 2: 14 * 7 = 98. Step 3: 100 - 98 = 2. So 100 = 14 * 7 + 2, and the round trip checks: 14 * 7 + 2 = 100.
Now a = -17 with b = 5. Since -17 / 5 = -3.4, floor gives -4, not -3. Then -4 * 5 = -20 and -17 - (-20) = 3, so -17 = -4 * 5 + 3. Truncation toward zero (the rule in C and Java) would report -3 remainder -2 instead. Flooring is why a positive divisor always leaves a remainder of 0 or more.
Decimals take the same path: 7.5 / 2 floors to 3, then 3 * 2 = 6, then 7.5 - 6 = 1.5. The remainder 1.50 is correct, but the summary rounds the operands to whole numbers, so it reads 8 / 2 = 3 r 1.50.
Boundary cases: where the result stops being trustworthy
A zero divisor is rejected before any arithmetic runs. Negative divisors are allowed and they set the sign of the remainder: 17 / -5 gives quotient -4 and remainder -3, which verifies as -4 * -5 + (-3) = 17, while -17 / -5 gives quotient 3 and remainder -2, since 3 * -5 + (-2) = -17. The remainder carries the divisor's sign, so it is not a positive count when b < 0.
Precision stops at 2^53 = 9007199254740992. Type 9007199254740993 and JavaScript parses it as 9007199254740992, so the tool reports remainder 2 where exact arithmetic gives 3. Farther out the answer turns silently wrong: 1e21 / 7 returns remainder 0.00, yet 10^21 mod 7 = 6 exactly, and the printed quotient 142857142857142861824 overshoots the true 142857142857142857142 by 4682. Keep both inputs under 9007199254740992 for results you can trust.