Divisibility Rules

Divisibility Rules


1. Divisibility by 1


Rule: Every number is divisible by 1.


Ex:
• 25 ÷ 1 = 25
• 987 ÷ 1 = 987


2. Divisibility by 2



  1. Rule: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

  2. Ex: 124, 98, 460 are divisible by 2.


3. Divisibility by 3



  1. Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.

  2. Ex: 213 → 2 + 1 + 3 = 6 (divisible by 3) → so 213 is divisible by 3.


4. Divisibility by 4



  1. Rule: A number is divisible by 4 if its last two digits form a number divisible by 4.

  2. Ex: 912 → last two digits = 12 → 12 ÷ 4 = 3 → divisible by 4.


5. Divisibility by 5



  1. Rule: A number is divisible by 5 if it ends in 0 or 5.

  2. Ex: 45, 120, 315 are divisible by 5.


6. Divisibility by 6



  1. Rule: A number is divisible by 6 if it is divisible by both 2 and 3.

  2. Ex: 72 → even number (divisible by 2) and 7 + 2 = 9 (divisible by 3) → divisible by 6.


7. Divisibility by 7



  1. Rule: Double the last digit, subtract it from the rest of the number.
    If the result is divisible by 7, the number is also divisible by 7.

  2. Ex: 203 → 20 − (2×3) = 14 → 14 is divisible by 7 → 203 is divisible by 7.


8. Divisibility by 8



  1. Rule: A number is divisible by 8 if its last three digits form a number divisible by 8.

  2. Ex: 1232 → last three digits = 232 → 232 ÷ 8 = 29 → divisible by 8.


9. Divisibility by 9



  1. Rule: A number is divisible by 9 if the sum of all its digits is divisible by 9.

  2. Ex: 234 → 2 + 3 + 4 = 9 → divisible by 9.


10. Divisibility by 10



  1. Rule: A number is divisible by 10 if it ends in 0.

  2. Ex: 120, 340, 1000 are divisible by 10.


11. Divisibility by 11



  1. Rule: Find the difference between the sum of digits in odd places and even places.
    If the difference is 0 or divisible by 11, then the number is divisible by 11.

  2. Ex: 121 → (1 + 1) − 2 = 0 → divisible by 11.


12. Divisibility by 12



  1. Rule: A number is divisible by 12 if it is divisible by both 3 and 4.

  2. Ex: 132 → divisible by 3 (1+3+2=6) and by 4 (last two digits 32 ÷ 4=8) → divisible by 12.


13. Divisibility by 13



  1. Rule: Multiply the last digit by 9 and add it to the rest of the number.
    If the result is divisible by 13, then the number is divisible by 13.

  2. Ex: 104 → 10 + (9×4) = 46 → 46 is divisible by 13 → 104 is divisible by 13


14. Divisibility by 17


Rule: (Last digit × 5) − remaining number.


Ex:
• 289 → 28−(9×5) = −17
• 204 → 20−(4×5) = 0


15. Divisibility by 19


Rule: (Last digit × 2) + remaining number.


Ex:
• 114 → 11+(4×2) = 19
• 703 → 70+(3×2) = 76


16. Divisibility by 23


Rule: (Last digit × 7) + remaining number.


Ex:
• 414 → 41+(4×7) = 69
• 391 → 39+(1×7) = 46


  DIVISIBILITY RULES


     A. ÷ 2 { The last digit is even (0,2,4,6,8)}


     B. ÷ 3 {The sum of the digits is divisible by 3}


     C. + 4 { The last 2 digits are divisible by 4 }


     D. ÷ 5 { The last digit is a 0 or 5 } 


     E. ÷ 6 { The number is divisible by 2 and 3 }


     F. + 8 { The last 3 digits are divisible by 8 }


     G.  ÷ 9 { The sum of the digits is divisible by 9 }


     H.  ÷ 10 { The last digit is a zero }