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What is the greatest prime factor of 8^12 - 2^30?

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by Vincen » Wed Feb 21, 2018 2:02 am
What is the greatest prime factor of $$8^{12}-2^{30}?$$ A. 2
B. 3
C. 5
D. 7
E. 11

The OA is the option D.

Experts, may you show me how can I rewrite the expression and then get the prime factorization? I ask for your help. <i class="em em-disappointed"></i>
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by Brent@GMATPrepNow » Wed Feb 21, 2018 7:50 am
Vincen wrote:What is the greatest prime factor of $$8^{12}-2^{30}?$$ A. 2
B. 3
C. 5
D. 7
E. 11
To determine the greatest prime factor of 8^12 - 2^30, we must find the prime factorization of 8^12 - 2^30

Given: 8^12 - 2^30
Rewrite 8 as 2³ to get: (2³)^12 - 2^30
Apply Power of Power rule to get: 2^36 - 2^30
Factor out 2^30 to get: 2^30(2^6 - 1)
Evaluate: 2^30(64- 1)
Simplify: 2^30(63)
Factor 63 to get: (2^30)(3)(3)(7)

At this point, we can see that the greatest prime factor is 7

Answer: D

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Brent Hanneson - Creator of GMATPrepNow.com
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by Jeff@TargetTestPrep » Mon Feb 26, 2018 10:09 am
Vincen wrote:What is the greatest prime factor of $$8^{12}-2^{30}?$$ A. 2
B. 3
C. 5
D. 7
E. 11
We can simplify the given expression:

(2^3)^12 - 2^30

2^36 - 2^30

2^30(2^6 - 1) = 2^30(63) = 2^30 x 9 x 7 = 2^30 x 3^2 x 7

So the largest prime factor is 7.

Answer: D

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