Is positive integer k the square of an integer?

Question

1) k = p!, where p is a prime number
2) k + 1 = w2 - 2w + 2, where w > 0

IMPORTANT:  This question is beyond the scope of the GMAT. That said, most students will encounter GMAT math questions they can't answer (e.g., the above question). So, I'm curious about how you'd handle a question of this nature.

Share your thoughts!!

chetan2u · Answer

Is positive integer k the square of an integer?
 The question tests everything that a GMAT student should know

1) k = p!, where p is a prime number
Now factorial is the product of all positive integers till that number.
So p! =1*2*3..*p. This means prime number p will come only once in the product and hence the product cannot be a square.
Sufficient

2) k + 1 = w² - 2w + 2, where w > 0
So k = w^2-2w+1=(w-1)^2 ..
Now w is not given as an integer, so we cannot be sure if k is a square..
For example 
if w is an integer , Ans is YES.
But if w= \sqrt{3}+1 , then k=3 and Ans is NO..
Insufficient

A

ccooley · Answer

Interesting one. I'm not entirely sure that it's outside the scope of the GMAT, because you can solve it with just reasoning and basic math. That said, the reasoning is on the tricky side...

For  statement 1 , I'd argue that k  can't  be the square of an integer. In a square number, every prime factor is repeated an even number of times. For example, 6^2 = 36, and 36 = 2*2*3*3. 5^2 = 25 = 5*5. There won't be any 'solo' prime factors.

But if you take the factorial of a prime number, that prime number (p) will only appear once in the prime factorization of k. So, k can't be a square.

Statement 2 : Start by simplifying it with math. Here's what I get:

k + 1 = w² - 2w + 2, where w > 0
k = w² - 2w + 1
k = (w-1)²

This will be the square of an integer if w is an integer, but otherwise, it won't be. And since we don't know whether w is an integer, it's insufficient.

The correct answer is A.

BrentGMATPrepNow · Answer

You're absolutely right; it's not out of scope.

When I first created the question, statement 1 read "k = n!, where n is an integer greater than 1"
Then, as I was posting the question, I realized that making the number prime would require fewer words. In doing so, I made the question MUCH easier. 
My bad.

Cheers,
Brent

ocelot22 · Answer

IF YOU FIND MY SOLUTION HELPFUL PLEASE GIVE ME KUDOS

(1) k = p! where p is a prime number. This means that p = 2,3,5,7.... let p =2 k =2, so we get a no. Let p=3, then k = 3*2 = 6 so we get another no. Let p =5, then k =5*4*3*2*1 = 1*120, let p = 7. Then k = 7*6*5*4*3*2*1. The pattern that the test is trying to get you to see here, is that no matter what prime we pick, we will have a single leading prime factor, thus K can never be the square of an integer sufficient

(2) k+1 = w^2-2w+2 = (w-1)^2. Notice that there are no restrictions on w, or (w-1)^2 directly. Our only restriction is that k must be a positive integer.Then let (w-1)^2 = 0, (obtained when w=1)

This is an upper level 700 question for sure, but not nearly as bad as some of the 3 overlapping set problems. In my opinion, it is not unrealistic for an upper level test taker to run into a problem like this on test day.

Is positive integer k the square of an integer?

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Is positive integer k the square of an integer?

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