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

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What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 25 May 2016, 03:24
1
12
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A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

65% (01:08) correct 35% (01:21) wrong based on 320 sessions

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Status: Greatness begins beyond your comfort zone
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Re: What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 25 May 2016, 03:32
4
2
8^12 - 2^30
=(2^3)^12 - 2^30
= 2^36 - 2^30
= 2^30 (2^6 -1)
= 2^30 * (64-1)
= 2^30 * 63
= 2^30 * 9 *7
= 2^30 * 3^2 * 7
greatest prime factor is 7
Answer D
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Re: What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 25 May 2016, 07:50
1
Bunuel wrote:
What is the greatest prime factor of 8^12 - 2^30?

A. 2
B. 3
C. 5
D. 7
E. 11


8^12 - 2^30?

8 = 2^3
8^12 = 2^36

2^36 - 2^30 = 2^30 (2^6 - 1) = 2^30 (64 - 1) = 2^30 (63) = 2^30 (7*3*3)

hence 2,3,7 are the prime factors of 8^12 - 2^30. here, 7 is the greatest prime factor.

Option D is correct.
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What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 29 Dec 2016, 09:59
1
Bunuel wrote:
What is the greatest prime factor of 8^12 - 2^30?

A. 2
B. 3
C. 5
D. 7
E. 11


My take:

\(8\)\(^{12}\)\(-\)\(2^{30}\)

\(2\)\(^{3*12}\) \(- 2\)\(^{30}\)

\(2^{36}\) \(- 2^{30}\)

\(2^{30} (2^6-1)\)

\(2^{30} * 63\)

\(2^{30}*3*3*7\)

Hence "7" is the GREATEST prime (D)
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Re: What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 10 Apr 2017, 18:29
Bunuel wrote:
What is the greatest prime factor of 8^12 - 2^30?

A. 2
B. 3
C. 5
D. 7
E. 11


In order to solve this question we need to factor out the set of exponents until we find a prime number.

8^12-2^30
4^24-2^30
(2^24)( 2^12)-2^30
2^36-2^30
2^30-(2^6-1)
2^30-(64-1)
2^30-(63)

prime factorization of 63

(3)(3)(7)

Seven is the largest prime number and thus the answer
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Re: What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 21 Feb 2018, 07:52
1
Top Contributor
Bunuel 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 to get: ()^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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Re: What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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New post 22 Feb 2018, 16:48
Bunuel 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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Re: What is the greatest prime factor of 8^12 - 2^30? &nbs [#permalink] 22 Feb 2018, 16:48
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