Divides (Divisibility) — Definition, Formula & Examples

Divides means one integer goes into another with no remainder. When we say '3 divides 12,' we mean 12 ÷ 3 leaves zero remainder.

An integer $a$ divides an integer $b$ , written $a \mid b$ , if there exists an integer $k$ such that $b = a \cdot k$ . In this case, $b$ is divisible by $a$ .

Key Formula

a \mid b \iff b = a \cdot k \text{ for some integer } k

Where:

$a$ = The divisor (the number doing the dividing)
$b$ = The dividend (the number being divided)
$k$ = An integer quotient with no remainder

How It Works

The vertical bar symbol

\mid

is read as 'divides.' The number on the left is the potential divisor, and the number on the right is the number being divided. So

5 \mid 30

means '5 divides 30,' which is true because

30 = 5 \times 6

. If division does leave a remainder, we write

a \nmid b

, meaning '

a

does not divide

b

.' For example,

4 \nmid 10

because

10 ÷ 4 = 2

remainder

2

Worked Example

Problem: Determine whether 7 divides 42 and whether 7 divides 50.

Check 7 | 42: Find whether 42 equals 7 times some integer.

42 = 7 \times 6

Conclusion for 42: Since 6 is an integer, 7 divides 42.

7 \mid 42 \quad \checkmark

Check 7 | 50: Divide 50 by 7. You get 7 remainder 1, so no integer k satisfies 50 = 7k.

50 = 7 \times 7 + 1

Conclusion for 50: Since there is a remainder, 7 does not divide 50.

7 \nmid 50

Answer:

7 \mid 42

is true;

7 \nmid 50

Why It Matters

Divisibility is the foundation for finding factors, simplifying fractions, and computing the GCF or LCM of two numbers. You will also rely on it heavily when learning about prime factorization and solving problems in number theory.

Common Mistakes

Mistake: Reading

a \mid b

backwards, thinking it means 'a is divisible by b.'

Correction: The divider is on the left.

a \mid b

means 'a divides b,' so b is divisible by a. Think: the bar points from the smaller role (divisor) toward the larger role (multiple).