Divisibility Rules

1. Divisibility by 2:
   • A number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8).
   • Example: 128 is divisible by 2 because the last digit is 8.
2. Divisibility by 3:
   • A number is divisible by 3 if the sum of its digits is divisible by 3.
   • Example: 123 is divisible by 3 because 1+2+3=61 + 2 + 3 = 61+2+3=6, and 6 is divisible by 3.
3. Divisibility by 5:
   • A number is divisible by 5 if the last digit is either 0 or 5.
   • Example: 145 is divisible by 5 because the last digit is 5.
4. Divisibility by 9:
   • A number is divisible by 9 if the sum of its digits is divisible by 9.
   • Example: 729 is divisible by 9 because 7+2+9=187 + 2 + 9 = 187+2+9=18, and 18 is divisible by 9.
5. Divisibility by 11:
   • A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or divisible by 11.
   • Example: 121 is divisible by 11 because (1+1)−2=0(1 + 1) – 2 = 0(1+1)−2=0, and 0 is divisible by 11.

Examples :

Divisibility by 2:

• a) 452 – Yes (last digit 2)
• b) 379 – No (last digit 9)
• c) 860 – Yes (last digit 0)
• d) 1349 – No (last digit 9)

Divisibility by 3:

• a) 321 – Yes (3+2+1=63 + 2 + 1 = 63+2+1=6, 6 is divisible by 3)
• b) 745 – No (7+4+5=167 + 4 + 5 = 167+4+5=16, 16 is not divisible by 3)
• c) 912 – Yes (9+1+2=129 + 1 + 2 = 129+1+2=12, 12 is divisible by 3)
• d) 484 – No (4+8+4=164 + 8 + 4 = 164+8+4=16, 16 is not divisible by 3)

Divisibility by 5:

• a) 135 – Yes (last digit 5)
• b) 222 – No (last digit 2)
• c) 405 – Yes (last digit 5)
• d) 6780 – Yes (last digit 0)

Divisibility by 9:

• a) 243 – Yes (2+4+3=92 + 4 + 3 = 92+4+3=9, 9 is divisible by 9)
• b) 546 – Yes (5+4+6=155 + 4 + 6 = 155+4+6=15, 15 is divisible by 9)
• c) 801 – No (8+0+1=98 + 0 + 1 = 98+0+1=9, 9 is divisible by 9)
• d) 3762 – Yes (3+7+6+2=183 + 7 + 6 + 2 = 183+7+6+2=18, 18 is divisible by 9)

Divisibility by 11:

• a) 253 – No ((2+3)−5=0(2 + 3) – 5 = 0(2+3)−5=0, 0 is divisible by 11)
• b) 1331 – Yes ((1+3)−(3+1)=0(1 + 3) – (3 + 1) = 0(1+3)−(3+1)=0, 0 is divisible by 11)
• c) 2024 – Yes ((2+4)−0=6(2 + 4) – 0 = 6(2+4)−0=6, 6 is not divisible by 11)
• d) 341 – No ((3+1)−4=0(3 + 1) – 4 = 0(3+1)−4=0, 0 is divisible by 11)

Divisibility by 4:

• Rule: A number is divisible by 4 if the last two digits of the number form a number divisible by 4.
• Example: 7328 is divisible by 4 because 28 (the last two digits) is divisible by 4.

Divisibility by 6:

• Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
• Example: 432 is divisible by 6 because it is divisible by 2 (last digit is 2) and by 3 (sum of digits 4+3+2=94 + 3 + 2 = 94+3+2=9 is divisible by 3).

Divisibility by 7:

• Rule: A bit more complex, but one method is to double the last digit, subtract it from the rest of the number, and if the result is divisible by 7, then the original number is also divisible by 7.
• Example: 483. Double the last digit (3 ×\times× 2 = 6), subtract from the rest of the number (48 – 6 = 42). Since 42 is divisible by 7, 483 is also divisible by 7.

Divisibility by 8:

• Rule: A number is divisible by 8 if the last three digits of the number form a number divisible by 8.
• Example: 5616 is divisible by 8 because 616 (the last three digits) is divisible by 8.

Divisibility by 10:

• Rule: A number is divisible by 10 if its last digit is 0.
• Example: 12340 is divisible by 10 because the last digit is 0.

Divisibility by 12:

• Rule: A number is divisible by 12 if it is divisible by both 3 and 4.
• Example: 144 is divisible by 12 because it is divisible by 3 (sum of digits 1+4+4=91 + 4 + 4 = 91+4+4=9 is divisible by 3) and by 4 (last two digits 44 are divisible by 4).

Divisibility by 13:

• Rule: A method is to remove the last digit, multiply it by 9, and subtract from the rest of the number. If the result is divisible by 13, then the original number is divisible by 13.
• Example: For 273, remove the last digit (3), multiply by 9 (3 ×\times× 9 = 27), and subtract from the rest of the number (27 – 27 = 0). Since 0 is divisible by 13, 273 is divisible by 13.

Divisibility by 15:

• Rule: A number is divisible by 15 if it is divisible by both 3 and 5.
• Example: 390 is divisible by 15 because it is divisible by 3 (sum of digits 3+9+0=123 + 9 + 0 = 123+9+0=12 is divisible by 3) and by 5 (last digit is 0).

Divisibility by 17:

• Rule: Remove the last digit, multiply it by 5, and subtract from the rest of the number. If the result is divisible by 17, then the original number is divisible by 17.
• Example: For 289, remove the last digit (9), multiply by 5 (9 ×\times× 5 = 45), and subtract from the rest of the number (28 – 45 = -17). Since -17 is divisible by 17, 289 is divisible by 17.

Divisibility by 18:

• Rule: A number is divisible by 18 if it is divisible by both 2 and 9.
• Example: 324 is divisible by 18 because it is divisible by 2 (last digit is 4) and by 9 (sum of digits 3+2+4=93 + 2 + 4 = 93+2+4=9 is divisible by 9).

Divisibility by 25:

• Rule: A number is divisible by 25 if the last two digits are 00, 25, 50, or 75.
• Example: 4250 is divisible by 25 because the last two digits are 50.

Divisibility by 100:

• Rule: A number is divisible by 100 if the last two digits are 00.
• Example: 3400 is divisible by 100 because the last two digits are 00.

Examples :

1. Determine if the following numbers are divisible by 4:

• a) 132
  • Last two digits: 32
  • 32 ÷ 4 = 8 (exact division, no remainder)
  • Result: Yes, 132 is divisible by 4.
• b) 576
  • Last two digits: 76
  • 76 ÷ 4 = 19 (exact division, no remainder)
  • Result: Yes, 576 is divisible by 4.
• c) 891
  • Last two digits: 91
  • 91 ÷ 4 = 22.75 (not an exact division)
  • Result: No, 891 is not divisible by 4.

2. Check divisibility by 6 for the following numbers:

• a) 234
  • Check divisibility by 2: Last digit is 4 (even)
  • Check divisibility by 3: 2+3+4=92 + 3 + 4 = 92+3+4=9 (9 is divisible by 3)
  • Result: Yes, 234 is divisible by 6.
• b) 455
  • Check divisibility by 2: Last digit is 5 (odd, not divisible by 2)
  • Result: No, 455 is not divisible by 6.
• c) 672
  • Check divisibility by 2: Last digit is 2 (even)
  • Check divisibility by 3: 6+7+2=156 + 7 + 2 = 156+7+2=15 (15 is divisible by 3)
  • Result: Yes, 672 is divisible by 6.

3. Test divisibility by 7 for these numbers:

• a) 329
  • Last digit: 9
  • Remove the last digit and subtract twice its value from the rest: 32−(9×2)=32−18=1432 – (9 \times 2) = 32 – 18 = 1432−(9×2)=32−18=14
  • 14 is divisible by 7.
  • Result: Yes, 329 is divisible by 7.
• b) 672
  • Last digit: 2
  • Remove the last digit and subtract twice its value from the rest: 67−(2×2)=67−4=6367 – (2 \times 2) = 67 – 4 = 6367−(2×2)=67−4=63
  • 63 is divisible by 7.
  • Result: Yes, 672 is divisible by 7.
• c) 889
  • Last digit: 9
  • Remove the last digit and subtract twice its value from the rest: 88−(9×2)=88−18=7088 – (9 \times 2) = 88 – 18 = 7088−(9×2)=88−18=70
  • 70 is divisible by 7.
  • Result: Yes, 889 is divisible by 7.

4. Identify if the following numbers are divisible by 8:

• a) 2048
  • Last three digits: 048
  • 048 ÷ 8 = 6 (exact division, no remainder)
  • Result: Yes, 2048 is divisible by 8.
• b) 4096
  • Last three digits: 096
  • 096 ÷ 8 = 12 (exact division, no remainder)
  • Result: Yes, 4096 is divisible by 8.
• c) 1256
  • Last three digits: 256
  • 256 ÷ 8 = 32 (exact division, no remainder)
  • Result: Yes, 1256 is divisible by 8.

5. Determine if the following numbers are divisible by 12:

• a) 276
  • Check divisibility by 3: 2+7+6=152 + 7 + 6 = 152+7+6=15 (15 is divisible by 3)
  • Check divisibility by 4: Last two digits 76, 76 ÷ 4 = 19 (exact division)
  • Result: Yes, 276 is divisible by 12.
• b) 432
  • Check divisibility by 3: 4+3+2=94 + 3 + 2 = 94+3+2=9 (9 is divisible by 3)
  • Check divisibility by 4: Last two digits 32, 32 ÷ 4 = 8 (exact division)
  • Result: Yes, 432 is divisible by 12.
• c) 589
  • Check divisibility by 3: 5+8+9=225 + 8 + 9 = 225+8+9=22 (22 is not divisible by 3)
  • Result: No, 589 is not divisible by 12.

6. Check divisibility by 25 for these numbers:

• a) 675
  • Last two digits: 75
  • Result: Yes, 675 is divisible by 25.
• b) 3200
  • Last two digits: 00
  • Result: Yes, 3200 is divisible by 25.
• c) 18050
  • Last two digits: 50
  • Result: Yes, 18050 is divisible by 25.

Short Forms

Divisibility by 2:

• Rule: Last digit is 0, 2, 4, 6, or 8.
• Example: 48 (last digit 8).

Divisibility by 3:

• Rule: Sum of digits is divisible by 3.
• Example: 123 (1+2+3=61 + 2 + 3 = 61+2+3=6).

Divisibility by 4:

• Rule: Last two digits form a number divisible by 4.
• Example: 132 (last two digits 32).

Divisibility by 5:

• Rule: Last digit is 0 or 5.
• Example: 145 (last digit 5).

Divisibility by 6:

• Rule: Divisible by both 2 and 3.
• Example: 234 (even last digit and sum 2+3+4=92 + 3 + 4 = 92+3+4=9).

Divisibility by 7:

• Rule: Double last digit, subtract from the rest of the number, result is divisible by 7.
• Example: 329 (32−2×9=1432 – 2 \times 9 = 1432−2×9=14).

Divisibility by 8:

• Rule: Last three digits form a number divisible by 8.
• Example: 2048 (last three digits 048).

Divisibility by 9:

• Rule: Sum of digits is divisible by 9.
• Example: 729 (7+2+9=187 + 2 + 9 = 187+2+9=18).

Divisibility by 10:

• Rule: Last digit is 0.
• Example: 470 (last digit 0).

Divisibility by 11:

• Rule: Difference between sum of digits in odd positions and even positions is divisible by 11.
• Example: 121 ((1+1)−2=0(1 + 1) – 2 = 0(1+1)−2=0).

Divisibility by 12:

• Rule: Divisible by both 3 and 4.
• Example: 432 (sum 4+3+2=94 + 3 + 2 = 94+3+2=9 and last two digits 32).

Divisibility by 13:

• Rule: Remove last digit, multiply by 9, subtract from the rest; if divisible by 13, original is too.
• Example: 273 (27−9×3=027 – 9 \times 3 = 027−9×3=0).

Divisibility by 15:

• Rule: Divisible by both 3 and 5.
• Example: 390 (sum 3+9+0=123 + 9 + 0 = 123+9+0=12 and last digit 0).

Divisibility by 17:

• Rule: Remove last digit, multiply by 5, subtract from the rest; if divisible by 17, original is too.
• Example: 289 (28−9×5=−1728 – 9 \times 5 = -1728−9×5=−17).

Divisibility by 18:

• Rule: Divisible by both 2 and 9.
• Example: 324 (even last digit and sum 3+2+4=93 + 2 + 4 = 93+2+4=9).

Divisibility by 25:

• Rule: Last two digits are 00, 25, 50, or 75.
• Example: 3200 (last two digits 00).

Divisibility by 100:

• Rule: Last two digits are 00.
• Example: 4500 (last two digits 00).