Is there anything you know of to help someone remember the important parts of prime factorization, like a special song or something?
Yes, there is at least one "song" that I know of to help a person remember some useful facts about prime factorization, and a quick Internet search will not only guide you toward a mnemonic device for that topic but will leave you no doubt that there are such devices for just about any mathematical topic you could possibly name -- from the simplest parts of the multiplication tables... to formulas used only for calculus (remember from previous posts that the term "mnemonic" is just a fancy word for any specific memory trick to aid in "clumping" separate pieces of information or retaining more information on a subject than you would be naturally inclined to. One type of mnemonic, for example, is a rhyme.) And, as you know, I am often an advocate of such memory tools because I have seen students benefit tremendously from using them. The complication in this case (prime factorization) is that the better "songs" seem to focus on prime numbers, which are important, but which is only one of the elements you would need to understand, practice, and master the process of prime factorization. Being able to distinguish prime numbers from numbers that are not prime is the type of thing that lawyers like to describe as 'necessary but not sufficient'. The point I'm building to is this: prime factorization is an important and useful area of math that is best learned without a collection of memory tricks; but there is good news: despite its complicated sounding name, prime factorization is a topic that is actually easier to remember from a true understanding than to remember from tricks.
Even better news is this: Now that I've gotten my plug for 'authentic learning and understanding' out of the way, there are indeed many little memory tricks to help you through the process. So here goes...
First, it helps to know what prime factorization actually is, and what purpose it serves. Answer: Prime factorization is the taking of a number and breaking it down into factors and then breaking those factors down into smaller and smaller factors until each and every factor you have remaining is what's called a "prime number."
So before we go into why one would want to go through this process, we need to review what a prime number is. And that is simple: A prime number is a number that has two distinct factors, and only two distinct factors, and those factors have to be:
(1) the number itself; and
(2) the number one.
So, how about the number one? Is that prime?
Because although the first factor (1) and the second factor (1) in the equation 1 x 1 = 1 are two factors, they are really the same number. So they are not 'distinct' (different from each other).
How about Zero? Is that a prime number?
0 x 5 = 0,
0 x 6 = 0,
0 x 7 = 0, etc... infinite, giving us far more than two factors.
What about 2?
2 x 1 = 2
Yes, 2 is the only even prime number there is.
Is the number 3 prime? Yes, 3 x 1 = 3, and there are no other whole numbers that would make factors of three to complete a similar equation.
4 x 1 = 4, but so does 2 x 2, which means there are at least three distinct factors: 4, 1, 2.
O.K. There are many reasons for prime factorization but here is one of the biggies: It is the easiest and most reliable way to find Lowest Common Denominators.
Here's how: When presented with two fractions with different denominators, you take the two denominators and "Prime factor" them on separate sides of a sheet of paper folded in half vertically.
Example: 5/24 and 7/36
Denominators: 24 and 36:
2 4 3 6
/ \ / \
8 3 9 4
/ \ / \ / \
2 4 3 3 2 2
Next: You pick each of the prime factors you see, one at a time, and determine which side has more of that particular factor. (I refer to that side as the "winning side"). Then you count up how many of that particular factor are on the winning side ONLY, and you copy down that number – not just once, but the number of times that it appear on the "winning side."
Here, the left side (24) has the greater number of 2's, and it has 3 of them so you write 2 2 2.
You are done with 2's. Next, you will notice another prime factor, a 3, and the right hand side (36) has the winning number with two 3's. You write down 3 3 next to 2 2 2, like this:
2 2 2 3 3
Finally, put multiplication signs between each number and multiply from left to right.
2 x 2 x 2 x 3 x 3 = 72.
So 72 is the lowest common denominator of 24 and 36. A couple more steps to make sure the new numerators are correct, and you're on your way to adding two fractions that had different denominators and might not have been so easy to do in your head!
Other than that, the most useful memory device for this area of math is what's known as the "divisibility tests". Often, when people learn the divisibility tests for the first time, they appear to have witnessed a miracle, and indeed these "tests" are extremely powerful tools and remarkably easy after just a few minutes of practice. But I will leave that portion of this discussion for another day. (Soon, though, I promise).
Hope this helps,