Modulo Calculator – Find the Remainder of Division

Calculate the modulo or remainder of any division instantly with our free online modulo calculator. Essential for programming, number theory, and cryptography applications.

Examples:

Dividend (a)

Divisor (n)

Understanding Modulo Operations

The modulo operation finds the remainder after division. When you divide a by n, you get a quotient and a remainder. The modulo operation returns just the remainder. For example, 17 / 5 = 3 remainder 2, so 17 mod 5 = 2.

Modulo is written as "a mod n" or "a % n" in programming. It's fundamental to computer science (array indexing, hash functions), cryptography (RSA, Diffie-Hellman), and everyday applications (checking if a number is even: n mod 2 = 0).

Modulo Formula

a mod n = a - n x floor(a/n)

Or equivalently: a = q x n + r, where r = a mod n and 0 ≤ r < n

Key Property

The result is always between 0 and n-1 (inclusive). This makes modulo perfect for "wrapping" values into a fixed range.

Negative Numbers

For negative dividends, we add n until the result is positive. -17 mod 5 = 3 (not -2).

Worked Examples

Example 1: Simple Modulo

17 mod 5

17 / 5 = 3.4, floor = 3

3 x 5 = 15

17 - 15 = 2

Answer: 2

Example 2: Even Division

20 mod 4

20 / 4 = 5 exactly

5 x 4 = 20

20 - 20 = 0

Answer: 0 (divides evenly)

Example 3: Negative Dividend

-17 mod 5

-17 / 5 = -3.4, floor = -4

-4 x 5 = -20

-17 - (-20) = 3

Answer: 3 (always positive!)

Example 4: Clock Arithmetic

25 hours on a 12-hour clock

25 mod 12

25 = 2 x 12 + 1

Answer: 1 o'clock

Quick Fact

The modulo operation is so important in computing that most programming languages have a dedicated operator for it (%). In Python, JavaScript, and C++, you write "a % n". The word "modulo" comes from Latin "modulus" meaning "measure" – it measures how far a number is from the nearest lower multiple.

Frequently Asked Questions

What's the difference between mod and remainder?

In mathematics, they're the same. In programming, the % operator may return negative results for negative inputs (called "truncated division"), while true modulo always returns positive. This calculator uses true mathematical modulo.

Can the divisor be negative?

Mathematically, we typically use positive divisors. If you need negative moduli, the sign convention varies. This calculator assumes positive divisors for consistent results.

What is modulo used for in programming?

Common uses include: checking even/odd (n % 2), wrapping array indices, implementing circular buffers, hash functions, generating random numbers in a range, and time calculations (seconds to minutes: s % 60).

How do I check if a number is divisible by n?

If a mod n = 0, then a is divisible by n. For example, 100 mod 5 = 0, so 100 is divisible by 5. This is faster than division for divisibility testing.

What does "congruent modulo n" mean?

Two numbers are congruent mod n if they have the same remainder. We write a ≡ b (mod n). For example, 17 ≡ 5 (mod 12) because both leave remainder 5 when divided by 12.

Why is my calculator showing a different result?

Some calculators and programming languages use truncated division for negative numbers, giving negative remainders. This calculator uses floored division, which always gives non-negative results – the mathematical standard.

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