GMAT Question of the Day – Problem Solving – Puzzle
Which of the following is the greatest prime factor of 74^2 – 47^2?
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.
Additional GMAT Exponents Puzzle Difference of Squares Practice
Here’s one of the toughest questions from the GMAT Official Guide. It’s exponents based and you’ll need difference of squares plus a little trick to change the format: (0.99999999/1.0001)−(0.99999991)/(1.0003) =
And another difficult GMAT Puzzle (a littles less challenging than the .99999999 question) with exponents and primes: The product of all the prime numbers less than 20 is closest to which of the following powers of 10?
Lastly, an exponents GMAT Question of the Day Puzzle.