Prime Factorization of 200 and 180
Prime Factorization of 200 and 180

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.

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