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Number Theory – Divisibility Rules Divisibility by 2 A number is divisible by 2 if its last digit is even. Even digits are: 0,2,4,6,8 Example: 348 is divisible by 2 because its last digit is 8. Divisibility by 3 A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 123 1+2+3=6 Since 6 is divisible by 3, 123 is divisible by 3 Divisibility by 4 A number is divisible by 4 if the last two digits form a number divisible by 4. Example: 1316 Last two digits: 16 Since 16 is divisible by 4, 1316 is divisible by 4 Divisibility by 5 A number is divisible by 5 if its last digit is: 0 or 5 Example: 125, 340, 675 Divisibility by 6 A number is divisible by 6 if it is divisible by both 2 and 3. Example: 132 Last digit =2, so divisible by 2. Digit sum: 1+3+2=6 So it is divisible by 3. Therefore, 132 is divisible by 6 Divisibility by 7 Take the last digit, multiply it by 2, and subtract the result from the remaining number. If the result is divisible by 7, the original number is divisible by 7. Example: 203 Last digit =3 20−(2×3)=20−6=14 Since 14 is divisible by 7, 203 is divisible by 7 Divisibility by 8 A number is divisible by 8 if its last three digits form a number divisible by 8. Example: 4128 Last three digits: 128 Since 128÷8=16, 4128 is divisible by 8 Divisibility by 9 A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 729 7+2+9=18 Since 18 is divisible by 9, 729 is divisible by 9 Divisibility by 10 A number is divisible by 10 if its last digit is: 0 Example: 450, 1200, 890 Divisibility by 11 Find the difference between the sum of digits in alternate positions. If the difference is 0 or a multiple of 11, the number is divisible by 11. Example: 121 (1+1)−2=0 Therefore, 121 is divisible by 11 Divisibility by 12 A number is divisible by 12 if it is divisible by both 3 and 4. Example: 312 Digit sum: 3+1+2=6 So it is divisible by 3. Last two digits: 12 So it is divisible by 4. Therefore, 312 is divisible by 12 Divisibility by 13 Multiply the last digit by 4 and add it to the remaining number. Repeat if necessary. If the result is divisible by 13, the original number is divisible by 13. Example: 286 28+(6×4)=28+24=52 Since 52 is divisible by 13, 286 is divisible by 13