## Thursday, March 27, 2014

### Divisibility rules

The ability to find factors of a number helps simplify division problems and increase the speed an accuracy of solving multiplication problems. Sometimes it's imperative to be able to factor a two or three digit number or higher in the middle of a multiplication problem. The rules in the upcoming paragraphs will help in this process. You could also quickly tell someone that 57,231 is divisible by 3 and 9. Simple rules will make people think you're a math genius, when in fact, any average person can learn these divisibility rules.

Let's start with the easier rules. To check if a number is divisible by 2 is to simply determine whether the number is even or odd. Any number ending in 0, 2, 4, 6, or 8, therefore any even number is divisible by two, any odd number isn't.

Also very simple is divisibility by 5. Any number ending in 0 or 5 is divisible by 5. This is obvious since the multiples of 5 (0, 5, 10, 15, 20, 25, 30, ....) all end in either 0 or 5.

Divisibility by 10 may be the simplest of all. Any number ending in 0 is divisible by 10. No other number is divisible by 10.

To see if a number is divisible by 4, check the last two digits of the number. If this two digit number is divisible by 4, the entire number is. For example, the number 16,232 is divisible by 4 since 32 is divisible by 4. If a number ends in 00, it is also divisible by 4 since 100, 200, 300, 400, 500, 600, 700, 800, and 900 are all divisible by 4.

Checking if a number is divisible by 8 is similar to the previous method for divisibility by 4. Except now we check the last 3 digits of the number. If the 3 digit number is divisible by 8, the entire number is. For example, 66,512 is divisible by 8 because 512 divided by 8 is 64. This works because 1,000 and any thousand is divisible by 8.

The test for divisibility by 3 is also quite simple. Add the digits of a number. If the sum is divisible by 3, the entire number is divisible by 3. In the example in the first paragraph, add the digits of 57,231. The sum of the numbers is 5 + 7 + 2 + 3 + 1 = 18. We know that 18 is divisible by 3, so 57,231 is divisible by 3. Likewise, the same kind of rule holds true for divisibility by 9. If the sum of the digits is divisible by 9, the entire number is divisible by 9. We know 18 is divisible by 9, so is 57,231.

Testing for divisibility of 6 is simple once we know how to determine if a number is divisible by 3. If a number is even and divisible by 3, then it's also divisible by 6. Therefore 3,168 is divisible by 6, but 2,115 is not.

Another interesting test is for divisibility by 11. Take the number you are testing and alternately subtract and add the digits. If the result is 0 or a multiple of 11, then the number is divisible by 11. For example, take 36,713 and apply the rule (3 - 6 + 7 -1 + 3 = 6). Therefore 36,713 is not divisible by 11. Now apply the rule to 619,091. We get 6 - 1 + 9 - 0 + 9 - 1 = 22. Since 22 is divisible by 11, so is 619,091.

Now we get to the most difficult divisibility rule. To test if a number is divisible by 7, add or subtract a number that is a multiple of 7 to the number you are checking. Try to add or subtract a multiple of 7 so the number you get ends in 0. For example, suppose you want to know if 6,358 is divisible by 7. If we add 42, which is a multiple of 7, we get 6,400. We can remove the 0's since we know dividing by 10 doesn't affect the divisibility of 7. Now we have 64. We know 64 is not divisible by 7, therefore 6,358 is not divisible by 7.

Let's try one more number to test for divisibility by 7. Suppose we have 1,106. If we add 14, we get 1,120. Drop the 0 to get 112. Now we can add 28 (another multiple of 7) to get 140. We know 140 is divisible by 7 and if not, we drop the 0 to get 14, which is clearly divisible by 7. Therefore 1,106 is divisible by 7.

I have taught many students these rules over the past 14 years, some come in handy, others are just very interesting bits of information to the math lover. For all practical purposes we can check if a number is divisible by another number by simply using our handy calculators. But for those interested in math, want to get better at math or simply want to impress someone with quick mental math, these divisibility rules will come in handy.