Remainder Calculator
Divide any two integers and instantly find the quotient and remainder. Results shown in R notation, mixed fraction, and decimal form with a verification step.
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How It Works
Divide the dividend by the divisor to get the quotient (rounded down to whole number). Multiply quotient by divisor and subtract from dividend to get the remainder.
Quotient = ⌊a ÷ b⌋, Remainder = a − (b × quotient)487 ÷ 32 = 15 R 7The fundamental relationship: dividend = divisor × quotient + remainder. This always holds true and is the key to verifying division results.
a = b × q + r32 × 15 + 7 = 487 ✓The result can be written as a mixed number: the quotient as the whole part and the remainder over the divisor as the fractional part.
a ÷ b = q + r/b487 ÷ 32 = 15 + 7/32 = 15 7/32The decimal form is simply the dividend divided by the divisor. Unlike integer division, this may be a non-terminating decimal.
Decimal = a / b487 ÷ 32 = 15.21875When dividend or divisor is negative, the quotient is negative if signs differ. The remainder takes the sign of the dividend in this calculator.
sign(q) = sign(a) × sign(b)−17 ÷ 5 = −3 R −2Remainder is used to determine if a number is even/odd (n % 2), to cycle through arrays, to check divisibility, and to solve scheduling problems.
n mod 2 = 0 → even, 1 → odd100 days from Monday: 100 mod 7 = 2 → WednesdayQuick Reference
Common examples — verify instantly above.
Simple
17 ÷ 5
3 R 2
Simple
100 ÷ 7
14 R 2
Simple
487 ÷ 32
15 R 7
Zero R
144 ÷ 12
12 R 0
Large
1000 ÷ 7
142 R 6
Small
5 ÷ 8
0 R 5
Large div
47 ÷ 47
1 R 0
Even/Odd
99 ÷ 2
49 R 1 (odd)
Tips & Shortcuts
To check if a number is divisible by n, compute the remainder. If remainder = 0, it is perfectly divisible.
A quick divisibility check: divisible by 2 if last digit is even; by 3 if digit sum is divisible by 3; by 5 if ends in 0 or 5.
The remainder is always less than the divisor. If you get a remainder ≥ divisor, recalculate the quotient.
In programming, the modulo operator (%) gives the remainder. Python: 17 % 5 = 2. JavaScript: 17 % 5 = 2.
To find what day of the week a date falls on, compute (days from reference) mod 7 and look up the result.
Remainder = 0 means the divisor is a factor of the dividend. This is useful for finding factors and checking primality.
Common Mistakes
Computing 17/5 = 3.4 and thinking the remainder is 0.4
The remainder is a whole number. 17 ÷ 5 = 3 R 2, not 3.4. Multiply 3×5=15, subtract from 17: remainder = 2.
Forgetting to verify the result
Always check: divisor × quotient + remainder = dividend. Example: 5×3+2=17 ✓. If this does not hold, recalculate.
Confusing quotient with remainder
Quotient = how many times divisor fits completely. Remainder = what is left over. In 17÷5: quotient=3, remainder=2.
Expecting remainder to be a decimal
Remainder is always a non-negative integer (for positive divisors). For decimal results, use the decimal output field.
Not handling zero remainder correctly
If remainder = 0, the division is exact. Write the answer as just the quotient with no R notation: 144 ÷ 12 = 12 (not 12 R 0).
Confusing remainder and modulo for negative numbers
For negative dividends, remainder keeps the sign of the dividend. Modulo keeps the sign of the divisor. Results differ for negatives.
Frequently Asked Questions
The remainder is what is left over after integer division. 17 ÷ 5 = 3 remainder 2, because 5 goes into 17 three times (15) with 2 left over.
Remainder = Dividend − (Divisor × Quotient). Or simply: r = a − b × floor(a/b).
For positive numbers they are the same. For negative numbers, remainder takes the sign of the dividend, while modulo takes the sign of the divisor.
No. The remainder is always between 0 and divisor−1 for positive divisors. If remainder ≥ divisor, the quotient was calculated incorrectly.
R stands for Remainder. 17 ÷ 5 = 3 R 2 means the quotient is 3 and the remainder is 2.
Use the formula: Divisor × Quotient + Remainder = Dividend. Example: 5 × 3 + 2 = 17 ✓
Modular arithmetic works with remainders — often called "clock arithmetic." 17 mod 5 = 2 means 17÷5 leaves remainder 2. It underlies: time (hours use mod 12 or mod 24), cryptography (RSA uses modular exponentiation), hash functions (hash(x) = x mod tableSize), and cyclic patterns in nature and engineering. Written 17 ≡ 2 (mod 5) — "17 is congruent to 2 modulo 5."
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