Factoring: Tricks to Make It Easy

Episode #3 of the course Foundations of mathematics by John Robin

 

Today, as we continue to build our foundation of mathematics, we will turn to one of the most fundamental things we can do with numbers: factoring.

You might recall this term from school. What I’m going to show you today is how factoring is much easier to do when you think about a number and how to decompose it, since every number that is not prime is composite, meaning it can be broken down into a string of prime numbers multiplied together.

 

Exponent Notation

First off, let’s develop notation to make our life easier. As an example, let’s work with 16. In the list from Lesson 1, recall that it “decomposed” into 2x2x2x2. If we did this with 32, we would find it decomposed into 2x2x2x2x2. This is getting cumbersome. 128 is 2x2x2x2x2x2x2. We run into the same problem as numbers get larger and larger. The number 288 is 2x2x2x2x2x3x3. Have I convinced you yet that we need an easier way to do this?

It will be helpful now to rewrite our list of the natural numbers 2 through 19:

2, 3, 2×2, 5, 2×3, 7, 2x2x2, 3×3, 2×5, 11, 2x2x3, 13, 2×7, 3×5, 2x2x2x2, 17, 2x3x3, 19, ….

Now I’m going to write out the list of numbers from 2 through 19 again, using an easier notation:

2, 3, 2², 5, 2(3), 7, 2³, 3², 2(5), 11, 2²(3), 13, 2(7), 3(5), 2⁴, 17, 2(3²), 19, …

I did two things:

1. I introduced something called an exponent, which you might also remember from school. In the case of 16, which was 2x2x2x2, the prime number 2 was multiplied four times, so we write 2⁴. The exponent is an instruction: It tells you, “Multiply this number by itself as many times as this number” (4, in our case).

2. I replaced the “x” with a bracket. When you see two numbers next to each other like this: a(b), you assume it means axb. 5(11) means 5 multiplied by 11. We can do this with several numbers multiplied together. For example, 2x3x5x11 would become 2(3)(5)(11). This is also called a product, i.e., a string of numbers multiplied together.

Putting this all together, 2x2x3x3x3x5x5x11x13x17x17 would become 2²(3³)(5²)(11)(13)(17²). This product multiplies to 111,582,900.

And looking at this in reverse: 111,582,900 is a composite number that can be broken down into a product of prime numbers: 2²(3³)(5²)(11)(13)(17²).

 

How to Factor Any Number

Take the number 6972. Always start by dividing the smallest prime, 2, as many times as possible:

6972 ÷ 2 = 2(3486)

2(3486) ÷ 2 = 2²(1743)

We can’t divide a third time by 2. If you try, you’ll end up with a decimal. So, we move onto the next prime, 3:

2²(1743) ÷ 3 = 2²(3)(581)

We can’t go any further with 3—same problem, we’ll get a decimal. If we try 5, we get the same problem. So, we move onto the next prime, 7:

2²(3)(581) ÷ 7 = 2²(3)(7)(83)

We can’t go any further with 7. You’ll find the same thing with the next primes: 11, 13, 17, 19, 23, 29, 31, and so on.

Quick trick: When you reach the late stage and have only a smaller two- to three-digit number at the end, take the square root of that number. For example, √83 = ~9.1. All primes greater than √83 will not divide, so you can stop at the closest one (here, it’s 7).

So just like that, we’ve found that 6972 = 2²(3)(7)(83), and we can form all the factors of this number by multiplying these primes together in any way we want, i.e., 4 (=2²), 6 (=2×3), 12 (=2²x3), 14 (=2×7), 21 (=3×7), 166 (=2×83), etc.

This will work with any number. You can divide by the primes in order of smallest to largest, then as your number on the right shrinks each time, take the square root to find out when you can stop.

Whenever you look at a number now, think about how you can break it down into a product of primes. As you’ll see tomorrow when we move on to fractions, approaching numbers like this will give you true power, especially when it comes to simplifying fractions.

 

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