Divisible — Definition, Formula & Examples

Divisible means a number can be divided by another number with no remainder. For example, 12 is divisible by 3 because 12 ÷ 3 = 4 exactly.

An integer $a$ is divisible by a nonzero integer $b$ if there exists an integer $q$ such that $a = b \times q$ . When this holds, we write $b \mid a$ and say that $b$ divides $a$ .

Key Formula

b \mid a \iff a = b \times q \text{ for some integer } q

Where:

$a$ = The number being divided (the dividend)
$b$ = The number dividing into a (the divisor)
$q$ = The quotient, which must be a whole number

How It Works

To check whether a number is divisible by another, divide and see if the remainder is zero. If

18 \div 6 = 3

with no remainder, then 18 is divisible by 6. If

18 \div 5 = 3

remainder

3

, then 18 is not divisible by 5. Quick divisibility rules can help: a number is divisible by 2 if its last digit is even, divisible by 3 if the sum of its digits is divisible by 3, and divisible by 5 if it ends in 0 or 5.

Worked Example

Problem: Is 36 divisible by 4?

Divide: Divide 36 by 4.

36 \div 4 = 9

Check the remainder: The result is 9 with no remainder. Since 9 is a whole number, 36 is divisible by 4.

36 = 4 \times 9

Answer: Yes, 36 is divisible by 4.

Why It Matters

Divisibility is the foundation for finding factors, simplifying fractions, and identifying prime numbers. You rely on it every time you reduce a fraction like

\frac{8}{12}

\frac{2}{3}

Common Mistakes

Mistake: Confusing the direction — saying "4 is divisible by 36" instead of "36 is divisible by 4."

Correction: The larger number is usually the one being divided. Ask: does 36 ÷ 4 give a whole number? Yes, so 36 is divisible by 4.