Divisibility Rules for 11 and 12

On Wednesday I explained how to determine if a number is divisible by 3, 6, and/or 9. I also posted a comment with a link to rules for 2, 3, 4, 5, 6, 7, 8, 9, and 10. Therefore, I think it is time for 11 and 12.

Divisibility Rule for 11

Let’s say that we want to determine if 31823 is evenly divisible by 11.  I will know that it is evenly divisible by 11 if 3 – 2 + 8 – 1 + 3 is evenly divisible by 11.  31823 is divisible by 11 because 3 – 2 + 8 – 1 + 3 is equal to 11.

Let’s try 9123:  3 – 2 + 1 – 9 = -7.  Since -7 is not divisible by 11, 9123 is not evenly divisible by 11.

Let’s try one more together to make sure you understand the pattern.  Is 9876543210 divisible by 11?  Since 0 – 1 + 2 – 3 + 4 – 5 + 6 – 7 + 8 – 9 = -5 and -5 is not evenly divisible by 11, 9876543210 is not evenly divisible by 11.

Okay, so now that we understand the pattern, how do we find a nearby number that is divisible by 11?  Believe it or not, you just have to subtract the number that you get from our math problem to the original number, to get one that is divisible by 11.

Let’s use 9123.  3 – 2 + 1 – 9 gave us -7.  Using that number, we can do the following:  9123 – (-7) = 9123 + 7 = 9130.  This means that 9130 is evenly divisible by 11.

Now let’s try our other number that wasn’t divisible by 11:  9876543210.  We will subtract -5 which came from 0 – 1 + 2 – 3 + 4 – 5 + 6 – 7 + 8 – 9.  Using that number, we will find the answer by doing this:  9876543210 – (-5) = 9876543210 + 5 = 9876543215.  That means that 9876543215 is evenly divisible by 11.

Divisibility Rule for 12

A number is divisible by 12 if it is divisible by both 3 and 4.  The reason this is true is because 12 = 3 × 4.  Knowing this, the only part that you may not understand is when a number is evenly divisible by 4.  This is the case when the last two digits of the number are divisible by four.

If you have trouble looking at a two digit number and figuring out if it is divisible by four, try the following:

1. If the right-most digit is odd, it is not divisible by four.
2. If the right-most digit is 0, 4, or 8, the next digit to the left must be even.
3. If the right-most digit is 2, or 6, the next digit to the left must be odd.

As I explained in my previous divisibility post, a number is evenly divisible by 3 only if the sum of the digits in the entire number is even divisible by three.