 # Prime Factors

- Aug 26, 04:00 AM Comments 

1,960 = 49 x 16 x y. What is the value of y?

When most testtakers encounter problems such as the one above, they have a predictable, yet incorrect, reaction. Most people fret when faced with large numbers that would take more than two minutes to deal with. But this reaction is the exact opposite of the reaction you should have when faced with a large number.

Always remember: the GMAT is a test of your critical thinking abilities. While basic arithmetic skills are a prerequisite for achieving a high score on test day, the GMAT is not particularly concerned with your ability to divide 1,960 by 49.

And, not only is the GMAT not concerned with your ability to divide 1,960 by 49, but it also rewards those who know that the GMAT does not want them to do such laborious arithmetic.

The reaction you should have to problems such as the one above is not, “wow this problem is going to take a long time,” but, “the GMAT does not expect me to take the time to do all that division; there must be a shortcut here.”

That shortcut is prime factorization. Any number can be expressed as a series of primes multiplied together. If we can determine those primes, we can avoid time-consuming math.

Let’s return to the problem above, but this time we will express all the numbers in terms of their prime factors. In order to start the process of determining these factors, just think of any two numbers that multiply to equal 1,960:

1,960 = 196 x 10

Neither of these numbers is prime, so we repeat the process:

196* x 10 = 14 x 14 x 2 x 5

2 and 5 are prime, so they stay as they are, but 14 is not, again break it down, repeating the process on both sides of the equation until only primes remain. After breaking your original equation down into prime factors, you end up with:

7 x 2 x 7 x 2 x 2 x 5 = 7 x 7 x 2 x 2 x 2 x y

Then just cancel out any numbers appearing on both sides, and you are left with:

y = 5

*quick side note: to save time on test day, make sure you know your squares up to 15!

Bret Ruber
Kaplan GMAT 