Take another look at 60. When you found the factor pairs of 60, you saw that it had 12 factors and 6 factor pairs.
From here on, pairs will be referred to as boring and interesting factor pairs. These are not technical terms, but the boring factor pair is the factor pair that involves 1 and the number itself. All other pairs are interesting pairs. Keep reading to see why!
Examine one of these factor pairs—4 × 15. One way to think about this pair is that 60 breaks down into 4 and 15. One way to express this relationship visually is to use a factor tree:
Now, the question arises—can you go further? Sure! Neither 4 nor 15 is prime, which means they both have factor pairs that you might find interesting. For example, 4 breaks down into 2 × 2, and 15 breaks down into 3 × 5:
Can you break it down any further? Not with interesting factor pairs. You could say that 2 = 2 × 1, for instance, but that doesn’t provide you any new information. The reason you can’t go any further is that 2, 2, 3, and 5 are all prime numbers. Prime numbers only have one boring factor pair. So when you find a prime factor, you will know that that branch of your factor tree has reached its end. You can go one step further and circle every prime number as you go, reminding you that you the branch can’t break down any further. The factor tree for 60 would look like this:
So after breaking down 60 into 4 and 15, and breaking 4 and 15 down, you end up with 60 equals 2 × 2 × 3 × 5.
What if you start with a different factor pair of 60? Create a factor tree for 60 in which the first breakdown you make is 6 × 10:
According to this factor tree 60 equals 2 × 3 × 2 × 5. Notice that, even though they’re in a different order, this is the same group of prime numbers as before. In fact, any way you break down 60, you will end up with the same prime factors: two 2’s, one 3, and one 5. Another way to say this is that 2 × 2 × 3 × 5 is the prime factorization of 60.
One way to think about prime factors is that they are the DNA of a number. Every number has a unique prime factorization. The only number that can be written as 2 × 2 × 3 × 5 is 60. Breaking down numbers into their prime factors is the key to answering many divisibility problems.
As you proceed through the chapter, pay special attention to what prime factors can tell you about a number and some different types of questions the GRE may ask. But because the prime factorization of a number is so important, first you need a fast, reliable way to find the prime factorization of any number.
A factor tree is the best way to find the prime factorization of a number. A number like 60 should be relatively straightforward to break down into primes, but what if you need the prime factorization of 630?
For large numbers, it’s often best to start with the smallest prime factors and work your way toward larger primes. This is why it’s good to know your divisibility rules.
Take a second to try on your own, then continue through the explanation.
Start by finding the smallest prime number that 630 is divisible by. The smallest prime number is 2. Because 630 is even, it must be divisible by 2: 630 ÷ 2 = 315. So your first breakdown of 630 is into 2 and 315:
Now you still need to factor 315. It’s not even, so it’s not divisible by 2. Is it divisible by 3? If the digits of 315 add up to a multiple of 3, it is. Because 3 + 1 + 5 = 9, which is a multiple of 3, then 315 is divisible by 3: 315 ÷ 3 = 105. Your factor tree now looks like this:
If 315 was not divisible by 2, then 105 won’t be either (the reason for this will be discussed later), but 105 might still be divisible by 3. Because 1 + 0 + 5 = 6, then 105 is divisible by 3: 105 ÷ 3 = 35. Your tree now looks like this:
Because 35 is not divisible by 3 (3 + 5 = 8, which is not a multiple of 3), the next number to try is 5. Because 35 ends in a 5, it is divisible by 5: 35 ÷ 5 = 7. Your tree now looks like this:
Every number on the tree has now been broken down as far as it can go. So the prime factorization of 630 is 2 × 3 × 3 × 5 × 7.
Alternatively, you could have split 630 into 63 and 10, because it’s easy to see that 630 is divisible by 10. Then you would proceed from there. Either way will get you to the same set of prime factors.
Now it’s time to get a little practice doing prime factorizations.
Find the prime factorization of 90.
Find the prime factorization of 72.
Find the prime factorization of 105.
Find the prime factorization of 120.