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

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Joined: 02 Sep 2009
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What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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25 May 2016, 04:24
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64% (01:04) correct 36% (01:16) wrong based on 317 sessions

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

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

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

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25 May 2016, 04:32
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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
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Re: What is the greatest prime factor of 8^12 - 2^30?  [#permalink]

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25 May 2016, 08:50
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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29 Dec 2016, 10:59
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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10 Apr 2017, 19:29
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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21 Feb 2018, 08:52
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

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

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22 Feb 2018, 17:48
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.

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

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07 Mar 2019, 08:11
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Re: What is the greatest prime factor of 8^12 - 2^30?   [#permalink] 07 Mar 2019, 08:11
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