Okay. I wrote a post all about a math problem.

Really. The whole thing.

You've been warned. If you don't want to read a lengthy post about 6th grade math, you should go look in the archives.

I have a new obsession.

Perhaps I missed this trick in middle school,

or (as I suspect) I was never shown it,

but I've found this pair of techniques to be

fascinating and (dare I say it) elegant.

In my continuing exploration of middle

school math both to be a better tutor and

because I find it addictive (everyone has hobbies),

I've been playing with prime factorization.

We called them Factor Trees in my day. You know,

these things:

My own ancient memories of them were that they

were pointless seeming exercises in breaking down

numbers into their bits and pieces. But in this here

modern world of math, they do things with them.

This first trick, I vaguely remember.

Start by rewriting the factors like this:

If you take all the prime factors both numbers have

in common (two 2's and a 3), and multiply them out,

you have 12, which is the largest number that can

divide into both numbers evenly, the Greatest

Common Factor.

Okay, the second trick is for finding the Least Common

Multiple. It's the number both numbers (180 and 48)

can divide into. I remember writing long lists of

multiples until I would stumble upon a number they

had in common, or just multiplying both numbers

together and dealing with HUGE numbers.

But this is a slick, pretty, and (dare I say it) easy

solution. Look at the prime factors again,

This time you take all of the factors you

need to make both numbers. Think of this

as a building a kit to build two different

lego shapes. You can share blocks, but you

have to have all the right shapes.

So, we need four 2's, two 3's, and a 5 to

make both numbers. If you multiply

these numbers, you get 720,

which is the smallest number both 180

and 48 divide into evenly.

Why you ask? (Okay, why I ask.)

The four 2's, two 3's, and a 5,

can be combined like this:

(2 x 2 x 2 x 2 x 3) x 3 x 5 = 48 x 15

or like this:

(2 x 2 x 3 x 3 x 5) x 2 x 2= 180 x 4.

I don't know why I find this so cool, but I do.