An introduction to divisibility rules

In this article I will discuss how you can quickly tell if an integer is divisible by 7, 11, or 12. The tricks for 7 and 11 are somewhat more complicated than the tricks for a lot of the other small positive integers. However, in math more complicated can often mean more exciting and interesting as well. Additionally, once you get the hang of these tricks you will feel great for having accomplished something that is not extremely easy.

How to tell if an integer can be divided by 7!

Take the last digit of the number and multiply it by 2. Then, subtract your new number from the remaining digits. If your final number is divisible by 7, then the original number is also divisible by 7. Don’t be scared if you have to work with negative numbers! For this situation you can just ignore the negative sign. Just like with 3 and 9 you might have to repeat this process.

Examples:

49 can be divided by 7 because 9 * 2 = 18 and 4 – 18 = - 14. – 14 is divisible by 7 thus 49 is divisible by 7.

What about 4582? Multiplying the last digit (2) by 2 we get 4, then we subtract: 458 – 4 = 454. It might be difficult to tell if 7 is a factor of 454 so we repeat the process. 4 * 2 = 8 and 45 – 8 = 37 which certainly cannot be divided by 7, therefore 4582 is not divisible by 7.

How to tell if an integer can be divided by 11!

This is my favorite of all of the divisibility tricks. Starting with the left-most digit count off by 2’s.

1 … 2 … 1 … 2 … etc.

Add up all the 1’s then add up all the 2’s separately. Subtract the sum of the 1’s from the sum of the 2’s (or vice versa). If 11 is a factor of your new number, then 11 is a factor of the original number. Don’t forget that 11 is a factor of 0 because 11 * 0 = 0.

Examples:

1320: 3 + 0 = 3 and 1 + 2 = 3. 3 – 3 = 0 and 0 is divisible by 11 so 1320 is divisible by 11.

83492: 3 + 9 = 12 and 8 + 4 + 2 = 14. 14 – 12 = 2 and 2 is not divisible by 11 so 83492 is not divisible by 11.

How to tell if an integer is divisible by 12!

Just like you can break 6 into 2 and 3 you can also break down 12 into 4 and 3. If a number is divisible by 3 and it is divisible by 4, then it is divisible by 12.

Texas Instruments TI-30X IIS 2-Line Scientific Calculator, Black
Amazon Price: \$21.95 \$7.99 Buy Now
(price as of Jul 30, 2014)