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Factoring Tricks August 29, 2009

Posted by Ms. Miller in Algebra I, Geometry.

It’s important, if you are trying to find the LCM of some numbers (such as trying to find a common denominator), that you be able to tell immediately what factors go into a number. My 6th grade teacher, Mrs. Dana, taught me the following tricks, which I want to share with you:

“Casting Out Nines”
This is the name of a process of adding up the digits of a number; it will come in handy as we try to find factors. When we talk about “adding up the digits of a number”, what you should do is:

  • Cross out any 9s that are in the number.
  • If there are any pairs of digits that add up to 9, cross them out as well.
  • Add up the remaining digits. If the sum is greater than 9, continue to add the digits together until you have a one-digit answer. This is the number we are interested in.

Example: What is the sum of the digits of the number 96,216?
We cross off the 9, and then add 6+2+1+6=15. We then find the sum of 15, which is 1+5=6. So the sum of the digits is 6.

(This method also has uses in checking arithmetic, but we’re not concerned about that right now. If you’re interested in reading further about it, check out Wikipedia’s entry on the subject.)

Now, let’s look at how we know whether certain numbers are factors:

Factor Divisible By If …
1 Goes into everything
2 Number ends in 0, 2, 4, 6, or 8
3 The sum of the digits is 3, 6, or 9
4 The last two digits are divisible by 4. Ex. 87,665,9\underline{24}
5 Number ends in 0 or 5
6 Number is divisible by 2 and 3 (Even number whose digits sum to 3, 6, or 9)
7 No rule exists.
8 The last three digits are divisible by 8. Ex. 2783738273\underline{064}
9 The sum of the digits is 9.
10 Number ends in 0
12 Number is divisible by 3 and by 4.
25 Last two digits are divisible by 25 (00, 25, 50, 75).
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