What modulo means
Modulo is the remainder left after dividing. The result of a mod n is whatever is left over when a is divided by n:
a mod n = a − n × ⌊a ÷ n⌋
So 17 mod 5 = 2, because 5 goes into 17 three times (15) with 2 left over. The long division calculator shows the quotient and remainder side by side.
Negative numbers
Conventions differ. The mathematical modulo (shown as the main result) always returns a non-negative value for a positive modulus, so −17 mod 5 = 3. Most programming languages use a truncated remainder that keeps the sign of the dividend, giving −2 — shown as the third result for reference.
Where modulo shows up
- Even/odd: n mod 2 is 0 for even numbers, 1 for odd.
- Clock math: (hour + n) mod 12 wraps around the clock face.
- Cycling: index mod length loops back to the start of a list.
Frequently asked questions
What is the modulo operation?
Modulo gives the remainder after dividing one number by another. Written "a mod n", it is what is left over when a is divided by n. For example, 17 mod 5 = 2, because 17 divided by 5 is 3 with 2 left over.
How do you calculate a mod n?
Divide a by n, take the whole-number quotient, multiply it back, and subtract: a − (n × quotient). For 17 and 5, the quotient is 3, so 17 − (5 × 3) = 17 − 15 = 2.
How does modulo work with negative numbers?
It depends on the convention. The mathematical modulo always returns a non-negative result with a positive modulus, so −17 mod 5 = 3. Many programming languages instead return a result with the sign of the dividend (−2). This calculator shows the non-negative mathematical result and notes the truncated version.
What is modulo used for?
It is everywhere in computing and math — wrapping clock times (hours mod 12), checking even or odd (n mod 2), cycling through a list, hashing, and cryptography. Anytime something repeats in a cycle, modulo finds the position within that cycle.
What is the difference between modulo and division?
Division tells you how many whole times one number fits into another (the quotient); modulo tells you what is left over (the remainder). They are the two halves of the same operation — the long division calculator shows both together.
Disclaimer: Inputs are treated as whole numbers; a modulus of zero is undefined. Provided for educational purposes.