Modulo Calculator

Two conventions for negative numbers

For positive operands, all conventions agree: 17 mod 5 = 2. For negatives, languages differ:

Floored division: quotient rounds toward −∞. Remainder always has the sign of the divisor: same-sign as b. Truncated division: quotient rounds toward zero. Remainder has the sign of the dividend: same-sign as a.

For a = -17, b = 5:

• Truncated: q = -3 (rounded toward 0), r = -17 − (-3)·5 = -2.
• Floored: q = -4 (rounded toward -∞), r = -17 − (-4)·5 = 3.

Both are correct given the convention. Pick to match your language.

Where modulo shows up

• Time. Minutes in an hour, days of the week, seconds in a day, all mod arithmetic.
• Hash tables. hash(key) mod table_size picks a bucket.
• Round-robin scheduling. task_i mod worker_count assigns work.
• Cryptography. RSA and Diffie-Hellman are built on mod n with huge primes.
• Every Nth row. In SQL: WHERE id % 3 = 0.
• Even/odd check. n mod 2 == 0 means even.
• Alternating patterns. Row striping, coloring every other item.
• Circular buffers. (index + 1) mod size wraps around.

Useful identities

• (a + b) mod n = ((a mod n) + (b mod n)) mod n
• (a · b) mod n = ((a mod n) · (b mod n)) mod n
• (a^k) mod n can be computed by fast modular exponentiation in O(log k), critical for large-number crypto.
• a mod 1 = 0 for any integer a.
• a mod a = 0.

Decimal modulo

For real numbers, the natural definition is a mod b = a − b · floor(a/b). 7.5 mod 2.5 = 0 because 7.5 is an exact multiple. 7.6 mod 2.5 = 0.1.

JavaScript’s % operator works on real numbers; Python’s fmod does truncated real mod; Python’s % does floored real mod.

Worked examples

• 100 mod 7: 100 = 14·7 + 2, so remainder 2.
• 25 mod 4: 25 = 6·4 + 1, so remainder 1.
• -10 mod 3 (floored): -10 = -4·3 + 2, remainder 2. (Truncated would give -1.)
• 17.5 mod 5: 17.5 = 3·5 + 2.5, remainder 2.5.