Let’s say you have a large number, and you need to break it down into its prime factors (prime factors are prime numbers, e.g. 2, 3, 5, 7). When dealing with small numbers, such as 24 (2 x 2 x 2 x 3), finding the prime factors isn’t too tough. But what about 324?

Many students will freeze when they see such a number. However, on the GRE, there is always a way to break down a large number.

Divide by 2

Is the number even? If so, simply divide by 2. If that results in a number that is also divisible by 2 continue dividing until the number is no longer even. 324/2 = 162. 162/2 = 81. At this point, you should be able to break down 81 into 9 x 9. Next, 9 x 9 can further be broken down into 3 x 3 x 3 x 3. Therefore, the prime factors of 324 are 3, 3, 3, 3, 2, 2.

Now, let’s try another question:

What is the sum of the unique prime factors of 207?

(A) 23

(B) 26

(C) 29

(D) 69

(E) 207

Divide by 3

Well, 207 is a very unpleasant number. Is it even divisible by any number, besides 1? Well, a good rule when dealing with odd numbers is to add up the digits, e.g. 2 + 0 + 7 = 9. If the sum is divisible by 3, then the number is divisible by 3. Therefore, 207 is divisible by 3 (9 is divisible by 3). 207/3 = 69. What about 69? Add up the digits, and you get 15, which is also divisible by 3. Therefore, 69/3 = 23. 23 is a prime number because it can’t be divided by any number, except 1.

The factors of 207 are therefore 3, 3, and 23. Notice, the question asks for the sum of the unique factors. The factor 3 appears twice, so we can discount one of the threes (because it is not unique; there is already another 3). The answer is 23 + 3 = 26 (B).

Finally, let’s try one last question.

What is the range of the prime factors of x, where x is 275?

(A) 5

(B) 6

(C) 11

(D) 30

(E) 55

Divide by 5

If a number ends in a 5, it is always divisible by 5. 275/5 = 55. 55/5 = 11. Therefore, the prime factors are 5, 5, and 11. The range of these factors is the greatest number minus the smallest number. 11 -5 = 6. Answer (B).

**Takeaway**

Factoring/breaking down numbers is something you will need to do often on the GRE. Becoming adept at factoring quickly – either on paper or with a calculator – will save you a lot of time. So, practice as much as possible, and have more time test-day to walk through the really tough problems.

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Do we consider 1 as a factor of every number? If so then 1 is also an odd factor or not? I am a bit confused?

Hi Simran,

Yes, 1 is a factor of every number, and it is also an odd number. Hope that helps!

I have a doubt on finding no. of factors for any given number. For instance, we will take 4.

Factors are: 1,2 and 4; No. of factors: 3

No. of factors of number 4 is six if we include negative number -1, -2 and -4

Please enlighten me. I am confused totally between these two cases.

Hi Pandi,

You are correct that a positive number can have negative factor. However, the good news is that the GRE deals with just positive factors and multiples. So you don’t need to worry about negative factors and multiples for the GRE. So for the purposes of the GRE, 4 will only have three factors. 🙂

‘the new GRE is less than a month away’ line can be removed from this blog post – https://magoosh.com/gre/2011/gre-math-essential-tips-for-factoring/

Thanks for catching that 🙂

The range shouldn’t be inclusive of 5 and 11, i.e. if 7 was also in answer choices the correct answer would be 6 or 7?

Hi Abishek,

If the prime facts are 5 and 11, then the range would simply be the greatest number (11) minus the smallest number (5). If 7 were a factor, it is in the middle so it wouldn’t affect the range in this case. I’m not sure if I’m addressing your concern here. Let me know :).