Divisibility Rules: Concepts, Formulas and Examples

Divisibility rules are quick tests that tell you whether one number divides another without doing the full division. Instead of dividing 7,326 by 3 on paper, you add its digits — 7 + 3 + 2 + 6 = 18 — and since 18 is divisible by 3, so is 7,326. These shortcuts save time and catch calculation errors.

These rules matter because divisibility is the gateway to simplifying fractions, finding factors and testing primality. A student who can spot divisibility mentally works faster and makes fewer mistakes in every later topic.

Key Concepts

• Divisibility: a is divisible by b when a ÷ b leaves remainder 0, written b | a.
• Rule for 2: the last digit is even (0, 2, 4, 6, 8).
• Rule for 3: the sum of the digits is divisible by 3.
• Rule for 4: the last two digits form a number divisible by 4.
• Rule for 5: the last digit is 0 or 5. Rule for 10: the last digit is 0.
• Rule for 6: divisible by both 2 and 3. Rule for 9: the sum of the digits is divisible by 9.
• Rule for 8: the last three digits form a number divisible by 8. Rule for 11: the difference between the sum of digits in odd and even positions is 0 or divisible by 11.

Formulas

• Divisibility by 3: 3 | n exactly when 3 divides the sum of n's digits.
• Divisibility by 9: 9 | n exactly when 9 divides the sum of n's digits.
• Divisibility by 11: for n with digits d₁d₂…dₖ, compute (d₁ + d₃ + …) − (d₂ + d₄ + …); if the result is 0 or a multiple of 11, then 11 | n.
• Combined rules: 6 = 2 × 3, so test both; a number divisible by 4 is automatically divisible by 2.

Worked Example

Test whether 918,082 is divisible by 11. Write the digits with positions from the left: 9, 1, 8, 0, 8, 2. Sum of digits in odd positions (1st, 3rd, 5th): 9 + 8 + 8 = 25. Sum of digits in even positions (2nd, 4th, 6th): 1 + 0 + 2 = 3. Difference: 25 − 3 = 22, and 22 = 2 × 11, so 11 divides 918,082. Verify by division: 918,082 ÷ 11 = 83,462, and 83,462 × 11 = 918,082, confirming the rule. Quick check for 3 on 7,326: digit sum 7 + 3 + 2 + 6 = 18, and 18 ÷ 3 = 6, so 3 divides 7,326 (indeed 7,326 ÷ 3 = 2,442).

Common Misconceptions

• "The rules for 4 and 8 are the same." They differ: 4 uses the last two digits, 8 uses the last three.
• "Divisible by 6 means divisible by 2 or 3." It must be divisible by both: 9 is divisible by 3 but not by 2, so it is not divisible by 6.
• "Ending in 0 means divisible by everything." A number ending in 0 is divisible by 2, 5 and 10 — not by 3, 7 or 11 automatically.

Real-World Applications

• Simplifying fractions quickly: spotting that 84 and 126 share factors starts with divisibility tests.
• Checking long-division answers mentally before submitting.
• Packing problems: knowing whether 144 items split evenly into boxes of 6 or 8.

In short, divisibility rules turn slow division into instant digit checks. Learn the tests for 2, 3, 4, 5, 6, 8, 9, 10 and 11, practise them on real numbers, and always verify a surprising result with one actual division.