The remainder R is the amount left over after dividing an integer D by d: R = D − (d × ⌊D/d⌋). Always 0 ≤ R < d.

What is modular arithmetic?

D mod d is the remainder when D is divided by d. Used in computer science (clock arithmetic, hashing), cryptography, and checksums.

How do I check long division?

Multiply the divisor by the quotient and add the remainder: d × Q + R = D. If this equals the original dividend, the division is correct.

Long Division Calculator

Q: What is modular arithmetic?

D mod d is the remainder when D is divided by d. Used in computer science (clock arithmetic, hashing), cryptography, and checksums.

Q: How do I check long division?

Multiply the divisor by the quotient and add the remainder: d × Q + R = D. If this equals the original dividend, the division is correct.

Enter a dividend and a divisor to compute the integer quotient and remainder — the building blocks of long division — with decimal equivalence and a verification formula.

How to use this calculator

Enter the dividend (the number being divided) and the divisor.
Read the integer quotient Q, remainder R, mixed number, and exact decimal.
Use the verification formula D = (d × Q) + R to confirm the result.

Formula & method

Integer division algorithm

Q = ⌊D / d⌋ · R = D − (d × Q) · Verification: D = (d × Q) + R

D = dividend, d = divisor, Q = quotient, R = remainder.

Example

487 ÷ 12: Q = ⌊487/12⌋ = 40, R = 487 − (12 × 40) = 7. Verification: 12 × 40 + 7 = 487 ✓

Key insights

Long division is the basis of polynomial division, Euclidean GCF algorithm, and modular arithmetic.
The remainder R is always 0 ≤ R < divisor. If R = 0, the dividend is exactly divisible.
Modular arithmetic uses the remainder: 487 mod 12 = 7 (remainder from this example).
The decimal result D / d can be expressed as the mixed number Q R/d for non-zero remainders.

How to interpret your result

Quotient and remainder

D = d × Q + R. The quotient Q is the number of complete groups; R is what is left over.

Mixed number

Q R/d expresses the result as a whole part plus a proper fraction. 487 ÷ 12 = 40 remainder 7 → 40 7/12.

Frequently asked questions

What is a remainder?: The remainder R is the amount left over after dividing an integer D by d: R = D − (d × ⌊D/d⌋). Always 0 ≤ R < d.
What is modular arithmetic?: D mod d is the remainder when D is divided by d. Used in computer science (clock arithmetic, hashing), cryptography, and checksums.
How do I check long division?: Multiply the divisor by the quotient and add the remainder: d × Q + R = D. If this equals the original dividend, the division is correct.

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