Source: Official Guide Revised GRE 1st Ed. Part 8; Section 6; #16


The integer v is greater than 1

The integer v is greater than 1. If v is the square of an integer, which of the following numbers must also be the square of an integer? Indicate all such numbers.

2 Explanations


Jorge Isaac Cordero Enriquez


There are some questions that ask "which ... *could* be..." and some others ask "which...*must* be...". As I understand it, when the question asks for a *could*, it is enough if we find a particular case that works, however, if the question asks for a *must*, then it must hold true for all cases.

This question asks for a *must*. How can we assure it holds true for all cases if we just tried with v=4?

Thanks :)

Oct 5, 2017 • Comment

Jonathan , Magoosh Tutor

Hi Jorge,
We prove algebraically that A and B must be true. 4 is just an example.
We know v is an integer from the question.
A) 81v = (9v)^2 and 9v must be an integer. So (A) must be correct.

B) We should know our FOIL identities cold, including (a + b)^2 = a^2 + 2ab + b^2

So We can see that (B) factors to

(5v + 1)^2 and we know 5v is an integer so therefore 5v + 1 is also an integer.

So (B) is correct.

For (C), we could see that we can't factor the expression so therefore there is no algebraic reason that it must be a square . To be sure, we can find one example for which it's not a square. If v = 4 The expression equals 75 which is not the square of an integer. So we know (C) is not correct.

Oct 30, 2017 • Reply


Gravatar Chris Lele, Magoosh Tutor

Oct 11, 2012 • Comment

Bethany D Zacharias

Thank you for all of these explanation videos. The GRE answer key is not nearly as easy to understand.

Oct 14, 2013 • Reply

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