Modulo — Definition, Formula & Examples

Modulo is an operation that gives you the remainder when one number is divided by another. For example, 17 modulo 5 equals 2, because 17 ÷ 5 = 3 with a remainder of 2.

Given integers $a$ and $n$ where $n > 0$ , the expression $a \bmod n$ yields the unique integer $r$ such that $a = qn + r$ and $0 \le r < n$ , where $q$ is an integer. The value $r$ is the remainder of the Euclidean division of $a$ by $n$ .

Key Formula

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

Where:

$a$ = The number being divided (the dividend)
$n$ = The number you divide by (the modulus), 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. If the division comes out evenly, the result is 0. For instance,

12 \bmod 4 = 0

because 4 goes into 12 exactly 3 times with nothing left over. You will often see the word "mod" written between two numbers, or the symbol

\%

used in programming languages to mean the same thing.

Worked Example

Problem: Find 23 mod 7.

Divide: Divide 23 by 7 to find the quotient.

23 \div 7 = 3 \text{ remainder } 2

Verify: Check by multiplying the quotient by 7 and adding the remainder.

3 \times 7 + 2 = 21 + 2 = 23 \checkmark

State the result: The remainder is 2, so the answer is 2.

23 \bmod 7 = 2

Answer:

23 \bmod 7 = 2

Why It Matters

Modulo is used constantly in computer science for tasks like determining whether a number is even or odd (

n \bmod 2

), cycling through lists, and cryptography. In everyday life, clock arithmetic is modulo 12 — fifteen hours after 1 o'clock lands on 4 o'clock because

15 \bmod 12 = 3

, and

1 + 3 = 4

Common Mistakes

Mistake: Confusing the modulo result with the quotient.

Correction: Modulo gives the remainder, not how many times the divisor fits. For

17 \bmod 5

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