Divisibility means a whole number can be divided by a natural number to get a whole number as the quotient with no remainder.

For example:

6÷3=2. The quotient 2 is a whole number. So we can say 6 is divisible by 3.

However 12÷5=2 ^{2}/_{5}. The quotient 2 ^{2}/_{5} is not a whole number so 12 is not divisible by 5.

Let’s get into some of the tricks now.

- Divisibility by 3 and 9
- Add all the digits of the number to get a new number. If that number is divisible by 3 or 9 then so is the original number.
- Example: Is 2168703 divisible by 3 and/or 9? 2+1+6+8+7+0+3=27. 27 is divisible by 3 (27/3=9). 27 is also divisible by 9. Therefore, 2168703 is divisible both by 3 and 9.

- Divisibility by 4 and 25
- Check the last 2 digits. If those 2 digits are divisible by 4 and/or 25 so is the original number.
- Example: Is 2168750 divisible by 4 and/or 25? The last 2 digits are 50. 50 is divisible by 25 (50/25=2). 50 is not divisible by 4 (50/4=25/2=12.5). Therefore, 2168750 is divisible by 25 but not by 9.

- Divisibility by 8 and 125
- Check the last 3 digits. If those 3 digits are divisible by 8 and/or 125 so is the original number.
- Example: Is 2168750 divisible by 4 and/or 25? The last 3 digits are 750. 750 is divisible by 125 (750/125=6). 750 is not divisible by 8 (750/8=93.75). Therefore, 2168750 is divisible by 125 but not by 8.

- Divisibility by 11
- Add all the even place digits and subtract from the sum of all the odd place digits. If the result is divisible by 11, so is the original number.
- Example: Is 5180956 divisible by 11? The sum of all of the even place digits is 1+0+5=6. The sum of all the odd place digits is 5+8+9+6=28. 28-6=22. 22 is divisible by 11. Therefore, 5180956 is divisible by 11.

- Divisibility by 7 and 13
- Subtract the last 3 digits from the remaining digits. If the result is divisible by 7 or 13 so is the original number. Repeat as needed.
- Example: Is 2168754 divisible by 7 and/or 13? The last 3 digits are 754. The remaing digis are 2168. 2168-754=1414. 1414 is divisible by 7 (1414/7=202). 1414 is not divisible by 13 (1414/13~108.8). Therefore, 2168754 is divisible by 7 but not by 13.

- Divisibility by 7 (another trick)
- Remove the last digit. Multiply that digit by 2 and subtract it from the remaining digits. If that number is divisible by 7, so is the original number.
- Example: Is 1414 divisible by 7? The last digit is 4. 2*4=8. The remaining digits are 141. 141-2*4=133. Do it again if you don’t know if 133 is divisible by 7. 13-2×3=7. 7 is divisible by 7 (7/7=1). Therefore, 1414 is divisible by 7.

Why don’t you check if these numbers are divisible by 3, 9, 4, 25, 8, 125, 7, 13, and 11.

- 5100
- 595
- 3861
- 5151575
- Try to find the divisibility of all of these with your 10 digit phone number 🙂

Leave your comments.

divisibility, math