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Difficulty:
25%
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Question Stats:
76% (01:23) correct
24%
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wrong
based on 1811
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What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
A. 2
B. 3
C. 7
D. 13
E. 17
01 Sep 2013, 22:00
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rrsnathan
Apply \(a^2-b^2=(a-b)(a+b)\).
\(3^6 - 1=(3^3-1)(3^3+1)=26*28=(2*13)*(2^2*7)\) --> the greatest prime factor is 13.
Answer: D.
23 Jan 2015, 14:47
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Hi All,
Both Smallwonder and Bunuel have provided elegant solutions to this question. The basic math behind this question is Arithmetic and Prime Factorization though, so if you don't immediately "see" the elegant approach, you can still get to the answer....
We're asked to find the LARGEST prime factor of 3^6 - 1
3^6 = 9^3 = (9)(9)(9) = 729
729 - 1 = 728
Now, we can prime factor 728.
You probably immediately see that 728 is divisible by 2, but if you know your 'rules of division', you can see that it's also divisible by 4...
728 =
(4)(182)
(4)(2)(91)
(4)(2)(7)(13)
13 is the largest prime.
Sometimes this type of approach isn't practical (especially if the numbers involved are HUGE), but when you're given 'manageable' numbers, there's nothing wrong with admitting that you don't see the 'hidden pattern.' If you can get to the correct answer in a reasonable amount of time by just doing arithmetic, then it's better to do THAT than waste time staring at the screen.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
Both Smallwonder and Bunuel have provided elegant solutions to this question. The basic math behind this question is Arithmetic and Prime Factorization though, so if you don't immediately "see" the elegant approach, you can still get to the answer....
We're asked to find the LARGEST prime factor of 3^6 - 1
3^6 = 9^3 = (9)(9)(9) = 729
729 - 1 = 728
Now, we can prime factor 728.
You probably immediately see that 728 is divisible by 2, but if you know your 'rules of division', you can see that it's also divisible by 4...
728 =
(4)(182)
(4)(2)(91)
(4)(2)(7)(13)
13 is the largest prime.
Sometimes this type of approach isn't practical (especially if the numbers involved are HUGE), but when you're given 'manageable' numbers, there's nothing wrong with admitting that you don't see the 'hidden pattern.' If you can get to the correct answer in a reasonable amount of time by just doing arithmetic, then it's better to do THAT than waste time staring at the screen.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
