Algebra Help: Expressing a Number as a Product of Prime Factors

An important skill the Algebra student needs to master early on is being able to express a number (specifically a counting number) as a product of prime factors. Before we continue, let's define a few terms. (This material is covered in Saxon Algebra 1, Lesson 33A).

The "counting numbers" is the set of whole numbers beginning with 1 and increasing by 1. So, the set = {1,2,3,4,....}. (As in no decimals allowed!)

A "product" is one or more numbers multiplied together. So for example 2*5 is a "product" that happens to equal 10. The individual numbers in the product are called "factors." (So the factors of 10 are 2 and 5.)

A "prime number" is a counting number greater than 1 whose only counting number factors are 1 and the number itself. For example, 11 is a prime number because 1*11 = 11. A "composite" number, therefore, is a counting number that has two counting number factors that are both greater than 1. For example, 8 is a composite number because 2*4 = 8.

A "prime factor" is simply a factor that is a prime number.

The algebra student should consider trying to memorize all prime numbers less than 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Knowing these primes can make doing algebraic calculations simpler in the future. You will note that with the exception of 2, primes are never even numbers. (Which makes sense because if they were, they would have 2 as a factor.)

Let's look at simple examples of expressing a counting number as a product of prime factors and work up to more complicated ones.

Express 8 as a product of prime factors:
Is 8 prime? No, but it is an even number, and therefore can be divided by 2:
8 = 2*4
Is 2 prime? Yes, but 4 isn't and therefore must be divided further:
8 = 2*2*2

(Many students won't need to be this methodical and will easily recognize that 8=2*2*2.)

Express 110 as a product of prime factors:
110 is clearly an even number and therefore divisible by 2.
110 = 2*55
We recognize that 55 is a multiple of the prime 11:
110 = 2*5*11

Express 69 as a product of prime factors:
69 is odd, so it is not divisible by 2, but it is divisible by 3:
69 = 3*23

Express 400 as a product of prime factors:
One could tackle this one from a variety of ways, depending upon the student's personality. I'll show two different sequences, either of which would provide the correct product:
400 = 4*100
400 = 2*2*2*50
400 = 2*2*2*2*25
400 = 2*2*2*2*5*5

Or one may decide to look at this way:
400 = 40*10
400 = 8*5*2*5
400 = 2*4*5*2*5
400 = 2*2*2*5*2*5

Normally the student is not required to list the factors in the product in any particular order, as long as he comes up with the correct list of primes.

Blessings!

Source
John H. Saxon, Jr. Algebra 1