Homework help 2 – prime factorization (elementary)

(The second in a semi-regular series. A math boot camp to help you help your kiddo, if you will.)

Recap: factors are numbers that go into your given.  So, given 40, my factors are 1, 2, 4, 5, 8, 10, 20 and 40.  Homework help 1 was all about finding those factors when the numbers weren’t so friendly.

New vocab:

  • Prime numbers are numbers that only have factors of 1 and themselves.  Like 17.  When I go to think of all the numbers that go into 17, I come back with nada.  Nothing goes into 17.  Well, nothing important.  I mean, 1 goes into everything.  And 17 goes into 17, but that’s kinda silly. As soon as I find myself thinking “well, nothing but 1 and 17 and that’s kinda silly” I know that 17 is prime.
  • Composite numbers are every other number.  Anything that’s not prime is composite.
  • 2 is the smallest prime number.  1 and 0 are just weirdos.  They don’t get to hang out at the prime number party.

We like prime numbers because they can’t break down any further.  If I asked you to break down 24, you could tell me that 24 is 2*12.  Or maybe 3*8.  Or get smarty pants and tell me 2*2*6.  Sometimes it’s handy for everyone to break down a number the same way.  To do this, we use prime factorization.

Lets do an example as a way of explaining.  If I asked you to break down 24, you might say that 24 = 2*12.  And I would say “you rock.”  (Because you do.)  But then I would ask you, are 2 and 12 prime?  You’d think for a minute and say that 2 is prime, but 12 is not, because 12=2*6.  So now we write 24 = 2*2*6.  Same question – are 2, 2, and 6 prime?  2 and 2 are, but 6 is not, because 6=2*3.  So now we write 24 = 2*2*2*3.  Sometimes people make this a picture:

See how each branch ends with a prime number?  And if we list the prime numbers in order, we get 2*2*2*3.

The power of prime factorization is that it TOTALLY DOESN’T MATTER what numbers you pick to divide by.  If you do the arithmetic right, you get the same answer in the end.  Let’s say you looked at 24, and thought 24=4*6.  Awesome!  Are either of those prime number?  Nope, because 4 = 2*2 and 6 = 2*3.  This means we can rewrite 24 = 2*2*2*3… The same answer as before!  Yay!  This factor tree picture would look like:

You can do factor trees for RIDICULOUSLY large numbers, fairly quickly if you’re good at times tables.

For example:

(Notice I circled the end of each branch when it got to a prime number, so I didn’t lose track of where my ends were.)

This means 960 = 2*5*2*2*3*2*2*2, or written in order 960 = 2*2*2*2*2*2*3*5

Not only is this an awesome conversation starter at a party (at least the parties I go too…), but it lets us reduce fractions in the next homework help post with almost no extra math at all.


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