Divisibility Rules — Definition, Rules & Examples

Divisibility rules are simple tests that tell you whether a number can be divided evenly by another number, without actually doing the division. For example, a quick rule tells you that any number ending in 0 or 5 is divisible by 5.

Divisibility rules are a set of shorthand criteria used to determine whether an integer is exactly divisible by a given divisor, meaning the remainder is zero. Each rule exploits a pattern in the number's digits — such as its last digit, the sum of its digits, or the alternating sum of its digits — to replace full division with a quick mental check.

Worked Example

Problem: Is 2,358 divisible by 2, 3, 4, 5, 6, and 9?

Divisible by 2?: Check whether the last digit is even (0, 2, 4, 6, or 8). The last digit is 8, which is even.

\text{Last digit} = 8 \quad \checkmark \text{ Yes}

Divisible by 3?: Add all the digits. If the sum is divisible by 3, so is the original number.

2 + 3 + 5 + 8 = 18 \quad \text{and } 18 \div 3 = 6 \quad \checkmark \text{ Yes}

Divisible by 4?: Look at the last two digits. If that two-digit number is divisible by 4, the whole number is too.

58 \div 4 = 14.5 \quad \text{✗ No}

Divisible by 5?: Check whether the last digit is 0 or 5. The last digit is 8.

\text{Last digit} = 8 \quad \text{✗ No}

Divisible by 6 and 9?: A number is divisible by 6 if it passes both the 2 and 3 rules — 2,358 does. For 9, the digit sum must be divisible by 9.

\text{By 6: passes 2 and 3} \quad \checkmark \text{ Yes} \qquad 18 \div 9 = 2 \quad \checkmark \text{ Yes}

Answer: 2,358 is divisible by 2, 3, 6, and 9, but not by 4 or 5.

Why It Matters

Divisibility rules save time whenever you need to simplify fractions, find factors, or test whether a number is prime. They're also useful outside the classroom — for example, quickly splitting a bill evenly or checking whether items can be packed into equal groups without leftovers.

Common Mistakes

Mistake: Confusing the rule for 3 with the rule for 4. Students sometimes add all the digits to test divisibility by 4.

Correction: The digit-sum trick works for 3 and 9 only. For 4, look at just the last two digits and check whether that two-digit number is divisible by 4.

Mistake: Forgetting that the rule for 6 requires both conditions (divisible by 2 AND by 3).

Correction: A number like 15 passes the test for 3 but fails for 2, so it is not divisible by 6. Always verify both rules.