Master Divisibility Rules from Zero

Divisibility rules are shortcuts to check divisibility WITHOUT doing actual division. Example: Is 12345678 divisible by 9? Instead of dividing, just add digits: 1+2+3+4+5+6+7+8 = 36 = divisible by 9. YES! These tricks save enormous time in MCQ exams.

Key Formulas

Divisibility by 2

Last digit is 0, 2, 4, 6 or 8 (even number)

Example: 456 is divisible by 2 (ends in 6).

Divisibility by 3

Sum of all digits is divisible by 3

Example: 123 → 1+2+3=6, divisible by 3 ✓

Divisibility by 4

Last 2 digits form a number divisible by 4

Example: 1236 → 36 ÷ 4 = 9 ✓

Divisibility by 5

Last digit is 0 or 5

Example: 235, 640 — both divisible by 5.

Divisibility by 6

Divisible by BOTH 2 and 3

Example: 48 — even (div by 2) and 4+8=12 (div by 3) ✓

Divisibility by 8

Last 3 digits form a number divisible by 8

Example: 1256 → 256 ÷ 8 = 32 ✓

Divisibility by 9

Sum of all digits is divisible by 9

Example: 729 → 7+2+9=18, divisible by 9 ✓

Divisibility by 11

Alternate digit sum difference is 0 or divisible by 11

Example: 121 → (1+1)-2=0 ✓. 1001 → (1+0)-(0+1)=0 ✓

⚡ Shortcut Tricks

Div by 7: No simple rule — use actual division. But remember: if N mod 7 = 0, it's divisible.

Div by 11 trick: (Sum of digits at ODD positions) - (Sum at EVEN positions) = 0 or multiple of 11.

Div by 12: Must be divisible by BOTH 3 and 4.

Div by 15: Must be divisible by BOTH 3 and 5.

Div by 18: Must be divisible by BOTH 2 and 9.

Div by composite number: Break into coprime factors. Example: 12 = 3 × 4 (coprime), check both.

✏️ Solved Examples

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