offers hundreds of GRE video lessons and practice questions. Go there now.
Sign up or log in to Magoosh GRE Prep.

Magoosh Brain Twister: The Terrible Threes – Explanation

brain_twister

As promised, here is the answer to Monday’s The Terrible Threes math Brain Twister!

First off, we should translate “what is the greatest value of x where n is an integer”. What the question is asking is how many 3’s can you divide out of 61! + 60! + 59!, or in other words how many factors of 3 does ‘n’ have.

Next, we’ll want to factor. The first step is to remove a 59! from each of the three numbers, since that is the biggest number in each: 59! [(61)(60) + (60) + 1].

Notice how (60) + 1 = 61. Notice as well that we already have ‘60’ 61’s (that’s what 61×60 means). By adding in the additional 61, we now have 61^2. Since ‘61’ is a prime, it has no factors of ‘3’, so we can focus our attention on 59!.

There are a lot of ‘3’s in 59!. First off, there are 19 multiples of ‘3’ (19×3 = 57). Secondly, there are 6 multiples of 9, which means an extra 6 factors of 3 (remember 9 = 3^2 and we already counted one of those threes when we found the multiples of ‘3’).

Finally, there are two multiples of 27 (27 and 54), which give us 2 additional factors of 3, since 27 = 3^3.

Therefore, 19 + 6 + 2 = 27.

 

About the Author

Chris Lele has been helping students excel on the GRE, GMAT, and SAT for the last 10 years. He is the Lead Content Developer and Tutor for Magoosh. His favorite food is wasabi-flavored almonds. Follow him on Google+!

One Response to Magoosh Brain Twister: The Terrible Threes – Explanation

  1. Cupcake July 27, 2014 at 9:29 am #

    Hey Chris,

    First of all: I hate your math problems. But then again, I can’t let go if I see a new one coming up every Monday ;)

    For this one, I didn’t really understand the part where it says:

    “There are a lot of ‘3’s in 59!. First off, there are 19 multiples of ‘3’ (19×3 = 57). Secondly, there are 6 multiples of 9, which means an extra 6 factors of 3 (remember 9 = 3^2 and we already counted one of those threes when we found the multiples of ‘3’).
    Finally, there are two multiples of 27 (27 and 54), which give us 2 additional factors of 3, since 27 = 3^3″

    How did you come up with the multiples? Did you went through every number and checked whether it is divisible by 3 or a multiple of 3? I find that rather exhausting – isn’t there a quicker way? Oh, and how likely is it that such questions appear on the GRE if you’re aiming for 165+ points?

    Thanks :)


Magoosh blog comment policy: To create the best experience for our readers, we will approve and respond to comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! :) If your comment was not approved, it likely did not adhere to these guidelines. If you are a Premium Magoosh student and would like more personalized service, you can use the Help tab on the Magoosh dashboard. Thanks!

Leave a Reply