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26 Nov 2014, 00:45
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
61% (02:31) correct
39%
(02:48)
wrong
based on 354
sessions
History
Date
Time
Result
Not Attempted Yet
If \(x = 8^6 - 5^6\), which of the following is NOT a factor of x?
(A) 39
(B) 43
(C) 47
(D) 49
(E) 63
(A) 39
(B) 43
(C) 47
(D) 49
(E) 63
26 Nov 2014, 16:32
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PareshGmat
Dear PareshGmat,
I'm happy to respond.
I think this question may lean on slightly more advanced math than the GMAT is likely to test. The GMAT will definitely hold students responsible for knowing the different of two squares pattern, and for knowing that something like x^6 is a square. Thus:
\(8^6 - 5^6 = (8^3 + 5^3)(8^3 - 5^3)\)
Now, at this point, the easiest way to simplify further would be sum of two cubes and difference of two cubes formulas, but I don't believe the GMAT expects test takers to have those formulas memorized. Incidentally, those formulas are
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
Alternately, we could simply compute the numerical values of the cubes. Yes, the GMAT would expect test takers to know that 5^3 = 125, but knowing that 8^3 = 512 is at the outer-edge gray-area of what the GMAT might expect. Suppose we go with that.
\(8^6 - 5^6 = (8^3 + 5^3)(8^3 - 5^3)\)
= (512 + 125)(512 - 125)
= (637)(387)
Well, 637 = 630 + 7, so it's clearly divisible by 7.
637 = 7*91 = 7*7*13
387 is divisible by 3 and 9. See:
https://magoosh.com/gmat/2012/gmat-divi ... shortcuts/
387 = 3*129 = 3*3*43
The prime factorization of the difference of sixth powers is 3*3*7*7*13*43
We can build (A) & (B) & (D) & (E) from these prime factors.
That leaves (C) as the answer. This is really at the very outer edge of something that the GMAT maybe could ask.
Mike
General Discussion
27 Nov 2014, 02:52
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\(8^6 - 5^6 = (8^3 + 5^3)(8^3 - 5^3)\)
\(= (8+5)(8^2 - 8*5 + 5^2)(8-5)(8^2 + 8*5 + 5^2)\)
\(= 13 * (64+25-40) * 3 (64+25+40)\)
\(= 13 * 49 * 3 * 129\)
\(= 49 * 39 * 43 * 3\)
Only 47 is not a factor
Answer = C
\(= (8+5)(8^2 - 8*5 + 5^2)(8-5)(8^2 + 8*5 + 5^2)\)
\(= 13 * (64+25-40) * 3 (64+25+40)\)
\(= 13 * 49 * 3 * 129\)
\(= 49 * 39 * 43 * 3\)
Only 47 is not a factor
Answer = C
