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# GMAT Question of the Day - Problem Solving - Puzzle

Which of the following is the greatest prime factor of $74^2$ - $47^2$?

A. 2

B. 3

C. 5

D. 11

E. 19

## GMAT Question of the Day Solution:

This is an exponent puzzle. The arithmetic would be possible but time consuming. It is rare that you would have to do so much arithmetic on a GMAT question so before you start chugging away take a pause and consider what other tools you have. With exponents there are a few things that you should consider:

1. Can you make the bases the same?

2. Can you factor?

3. Is it a special quadratic? (usually difference of squares)

In this case we have the difference of squares. You are probably used to seeing difference of squares in the format x^2 - y^2 but realize that there multiple ways that they can come up:

1. You could have a power other than two. Any even exponent will work so x^8 - y^12 is a difference of squares.

2. Real numbers also work so 74^2 - 47^2 is also a difference of squares.

3. 1 is a perfect square so 1 - y^8 would also be a difference of squares.

Get in the habit of converting difference of squares. This should be automatic. In this case, once you do that you can simplify each parenthesis so that you are only left with multiplication. Here's another place to use some GMAT sense: do not multiply out the parenthesis. We're looking for the greatest prime factor so guess what you should do: Prime Factorization. Whenever factors and multiples come up consider doing prime factorization. Make a factor tree pulling your primes out to the left (see below); this will keep your tree organized.

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