Modulo Operation — Definition, Formula & Examples

The modulo operation finds the remainder when one whole number is divided by another. For example, 17 mod 5 equals 2 because 17 ÷ 5 = 3 with a remainder of 2.

Given integers $a$ and $n$ with $n > 0$ , the expression $a \bmod n$ yields the unique integer $r$ such that $a = qn + r$ where $q$ is an integer and $0 \le r < n$ .

Key Formula

a \bmod n = r \quad \text{where} \quad a = qn + r \quad \text{and} \quad 0 \le r < n

Where:

$a$ = The number being divided (the dividend)
$n$ = The number you divide by (the divisor), must be positive
$q$ = The quotient — how many whole times n fits into a
$r$ = The remainder, which is the result of the modulo operation

How It Works

To compute

a \bmod n

, divide

a

n

and keep only the remainder. First, find how many times

n

fits into

a

without going over. Multiply that count by

n

, then subtract the result from

a

. What's left is the remainder. For instance,

23 \bmod 7

: since

7 \times 3 = 21

and

23 - 21 = 2

, the answer is 2.

Worked Example

Problem: Find 47 mod 6.

Divide: Divide 47 by 6 to find the whole-number quotient.

47 \div 6 = 7 \text{ remainder } ?

Multiply back: Multiply the quotient by the divisor.

7 \times 6 = 42

Subtract: Subtract to find the remainder.

47 - 42 = 5

Answer:

47 \bmod 6 = 5

Why It Matters

The modulo operation is essential in computer science for tasks like determining whether a number is even or odd (

n \bmod 2

), cycling through lists, and powering encryption algorithms. In everyday math, it helps with clock arithmetic — 15 hours after 10 o'clock lands on 1 o'clock because

(10 + 15) \bmod 12 = 1

Common Mistakes

Mistake: Confusing the modulo result with the quotient.

Correction: The modulo operation returns the remainder, not how many times the divisor fits in. For

17 \bmod 5

, the answer is 2 (the remainder), not 3 (the quotient).