In the last few blog posts we’ve explored “Invisible Gorillas”. As mentioned in the original post, Invisible Gorillas are patterns that recur on standardized tests that you can easily miss if you’re not looking for them. However, they’re simple to spot if you’re on the lookout for where they might show themselves.
I am not old. Granted, I didn’t get my first cell phone until I was in college. And yes, the first music I ever bought was on cassette tape. And sure, Google didn’t exist when I was born, but still. I am not being old. So why didn’t my teachers cover prime factorization when I was in school?
It’s not like the concept of prime factors is anything new. I would give you some historical evidence of this, but this newfangled Google thing gives me some trouble sometimes. But still, why wasn’t prime factorization more prevalent in schools when I was growing up?
For those of you as old as I am (or at least as disadvantaged as I was in not getting this information in school) prime factorization says that we can break any number down to smaller pieces—it’s prime bits—and make it easier to handle. We do that by first taking any factor pair and then seeing whether the factors are prime, or whether they are non-prime, in which case we repeat the process until we have all primes. Take the following example:
Our analysis shows us that 84=2*2*3*7. This in itself seems pretty simple, but making things simple is often the major step that you need to take in order to solve what looks like a complex problem. Take the following for example:
is a factor of 540. What is the greatest possible value of x+y?
Although the form makes this look daunting, it’s simply an exercise in prime factoring. Watch what happens when we re-write the information in that form:
So, 540= 2*2*3*3*3*5. All that’s left to do is re-write that in a form that more closely resembles the form in which the problem was given.
Once the number has been broken into its prime factors, it’s plain to see that we can only squeeze two twos and three threes out of this number. There’s no different way we could break it down to get more twos or threes. Prime factoring figures out how many pieces of each type we have and allows us to re-assemble them in any way we see fit. So the greatest possible x is 2 and the greatest possible y is 3. Thus, 2+3 is equal to our answer of five.
Whether you’re old or young or somewhere in between, jump on board with prime factorization because taking numbers and breaking them into their component prime parts is a great way to solve problems and a recurring pattern on your test that you’ll be sure to recognize if you look for it!