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x^3 – x^2 is a square

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by Brent@GMATPrepNow » Sat Feb 21, 2009 12:31 pm
If x is an integer, and 100≤x≤200, how many values of x exist such that x^3 – x^2 is a square?
A) 4
B) 5
C) 6
D) 7
E) 8

Please note that this is not an official GMAT question; it’s my attempt to create difficult (650+ level) GMAT-style questions for this forum.
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by DanaJ » Sat Feb 21, 2009 12:58 pm
Well, if x^3 - x^2 is a square, then x^2 (x - 1) is a square or x - 1 is a square. Since x is between 100 and 200, we have the following values for squares:
100
121
144
169
196
with corresponding +1 values. This makes for 5 solutions.

Welcome back, Brent! I have the feeling you haven;t posted much these past few days, but maybe I'm wrong... IMHO, your questions are one of the most representative for the GMAT....
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by Brent@GMATPrepNow » Sat Feb 21, 2009 1:02 pm
Nice work, Dana. The answer is 5
I've been working on other things, but will be posting more often again.
Thanks.
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by awesomeusername » Sat Feb 21, 2009 4:31 pm
It took me a bit to see that a square * a square = a square. Thus, if (x-1) is a square, then (x^2)(x-1) is a square. And like Dana said, there are 5 squares greater than or equal to 100 and less than or equal to 200.

Good question!
Constant dripping hollows out a stone.
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