## Ask Professor Puzzler

Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!

Fifth grader Aaradhya asks: "How do we find the factors of 342, 450 and 540 without a calculator?"

That's a really good question! The best suggestion I can give you is to make sure you are familiar with some divisibility rules. These are very helpful tools in breaking a number down into their factors.

For example, we know that any number that ends with an even digit is a multple of 2. So, right off the bat, we can divide 342 by 2. Of course, without a calculator you'll need to do long division. But then you've got smaller numbers to work with. In this case, 342 divided by 2 is 171.

Does that have any factors? Well, here's where the divisibility rule for 3 might come into play. A number is divisible by 3 if the sum of its digits is divisible by 3. Since 1 + 7 + 1 = 9, and 9 is divisible by 3, then we know 171 is divisible by 3 also. So, another quick long division and we've got 57. Once again, 5 + 7 is divisible by 3, so we know 57 is also. We continue like this until we get down to a number that is prime - in this case, 19. Thus, the factors of 342 are 1, 2, 3, 19, and 342.

There are other useful divisibility rules - for example, if a number ends with a zero, that means it's divisible by 10. Thus, 450 is 10 times 45. Then we just break these down iunto smaller and smaller factors.

Similar to this rule, if a number einds in zero or five, it's divisible by 5.

There are other divisibility rules, but they get increasingly complicated. The rules for seven and thirteen are so cumbersome that I never use them. The divisibility rule for 11 requires you to alternately add and subtract the digits to see if the sum is a multiple of eleven.

For example, 14641 is a multiple of 11 because 1 - 4 + 6 - 4 + 1 = 0 , which is divisible by 11.

Without the divisibility rules, this process requires a lot of long division just to find out if a number is a factor. You still may need to do some of that, but the divisibility rules will help you keep that to a minimum!