Modulo Calculator

What is the division remainder?

Euclidean division follows the relation dividend = divisor × quotient + remainder, with the condition 0 ≤ remainder < divisor. In code, the operator that hands back that remainder is modulo (%). Take 17 ÷ 5: the quotient is 3 and what is left over is 2.

You will run into this calculation everywhere, whether you are checking if a number is even or odd, handling cycles, building hash functions or even feeding cryptographic algorithms.

What the remainder is and the modulo operation

Integer division of a by b produces a quotient q = ⌊a / b⌋ and a remainder r = a - q · b, with 0 ≤ r < |b| in the mathematical convention. Dividing 17 by 5: 17 = 3 · 5 + 2, so quotient is 3 and remainder is 2. The remainder is also called the result of the modulo operation, written a mod b or a % b in most programming languages.

Negative operands expose a real gotcha: the sign of the remainder is not the same across languages. In C, C++, Java, JavaScript and Go, the result follows the sign of the dividend, so -7 % 3 = -1. In Python and Ruby, the result follows the sign of the divisor, so -7 % 3 = 2. Both satisfy a = q · b + r, but the floor versus truncated division choice differs. Always test edge cases when porting code between ecosystems.

Practical applications

• Parity test: n % 2 == 0 for even, != 0 for odd.
• Cycle wraparound: day of the week with days % 7, hours of the day with hours % 24.
• Hashing: distributing keys across N buckets with hash(key) % N.
• Public-key cryptography (RSA, Diffie-Hellman) and elliptic-curve schemes rely on modular arithmetic over very large primes.
• Check digits: CPF, CNPJ, barcodes (EAN-13), ISBN, IBAN and credit-card Luhn all use weighted sums modulo 10, 11 or 97.
• Bounded randomization: picking an index in an array via rand() % length.

FAQ

What happens if I divide by zero? Both quotient and remainder are undefined. Most languages raise an exception or return NaN; this tool blocks the operation and shows an error.

Why does Python give a different sign than JavaScript for -7 % 3? Python uses floor division (rounds toward minus infinity); JavaScript uses truncated division (rounds toward zero). Both keep the identity a = q · b + r, but the pair (q, r) is different.

Is mod the same as % in math notation? Almost. In number theory, a ≡ b (mod n) states a congruence relation rather than a single value, but the everyday programming use overlaps.

How do I get a strictly non-negative remainder? Use ((a % b) + b) % b — this normalizes the result to the range [0, b) regardless of the language convention.