## Ask Professor Puzzler

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"Are there any tricks that can help you easily factor three digit numbers (without using a calculator)?" ~Jay

Hi Jay, I assume you're talking about tricks besides the normal divisibility rules (for example, if the digits add to three, the number is a multiple of three, if it ends in 0 or 5 the number is divisble by 5, etc). If you're not familiar with those rules, you might want to take a look at this unit here: Divisibility Rules.

Beyond that, there are some tricks that *sometimes *help. Here's my favorite. Let's say you wanted to factor the number 483. Here's what I would do:

- Multiply the first and last digit: 4 x 3 = 12
- Find two numbers that multiply to 12 and add to the middle digit (8). The numbers are 6 and 2 (6 + 2 = 8 and 6 x 2 = 12).
- Now rewrite the number using those two numbers we just found: 483 = 460 + 23 (the tens place got split into two pieces using our numbers, and the entire number was rewritten as a sum of two numbers).
- Now factor the result: 460 + 23 = 23(20 + 1) = 23 x 21.
- Finish factoring: 23 x 7 x 3

Unfortunately, this doesn't always work. For example, it won't work for 648, because you can't find two numbers that add to 4 and multiply to 48. But maybe if we can find a way of regrouping this number, we might get around that. My first thought is to pull out one of the "hundreds" and put it into the tens place. So we're thinking of 648 as being rewritten 5(14)8. Now we do 5 x 8 = 40, and realize that our two numbers must be 4 and 10 (4 + 10 = 14 and 4 x 10 = 40). So we rewrite the number: (600 + 48 = 24(25 + 2) = 24 x 27. Then we just finish the prime factorization from there.

If the number is one of those special numbers (like 483) that can be factored without regrouping, it's a straightforward, foolproof process. But if the number has to be regrouped, it requires a bit of intuition to work it out. However, if you don't have a calculator, it might be worth doing!