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What is the largest prime factor of the expression 3^8 − 2^12?

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What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 10 Aug 2018, 02:24
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A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

44% (01:16) correct 56% (01:22) wrong based on 151 sessions

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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 10 Aug 2018, 02:30
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If you see it, it is in the (a+b)(a-b) which is = (3^4+2^6)(3^4-2^6) = (81+64)(81-64) = (145)(17)
Therefore, highest prime factor is 17.
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What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 10 Aug 2018, 02:32
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Bunuel wrote:
What is the largest prime factor of the expression \(3^8 − 2^{12}\)?

A. 2
B. 3
C. 5
D. 17
E. 29



Given

\(3^8 − 2^{12}\)

=\((3^4)^2 - (2^6)^2\)

=\(( 3^4 + 2^6) (3^4 - 2^6)\)

= (81 + 64) (81 - 64)

=\((145 ) ( 9^2 - 8^2)\)

= (5 * 29) ( 9 + 8) ( 9- 8)

= 5 * 29 * 17 * 1

The best answer is E.
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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 10 Aug 2018, 02:33
Bunuel wrote:
What is the largest prime factor of the expression 3^8 − 2^12?

A. 2
B. 3
C. 5
D. 17
E. 29


OA:E
\(3^8 − 2^{12} =(3^4)^2 − (2^6)^2 =(3^4-2^6)(3^4+2^6)=(3^2-2^3)(3^2+2^3)(3^4+2^6)=(9-8)(9+8)(81+64)=1*17*145=1*17*5*29\)
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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 10 Aug 2018, 02:33
Manick9 wrote:
If you see it, it is in the (a+b)(a-b) which is = (3^4+2^6)(3^4-2^6) = (81+64)(81-64) = (145)(17)
Therefore, highest prime factor is 17.



u can further break down 145

145 = 29 * 5

Hope it's clear.
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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 10 Aug 2018, 02:59
selim wrote:
Manick9 wrote:
If you see it, it is in the (a+b)(a-b) which is = (3^4+2^6)(3^4-2^6) = (81+64)(81-64) = (145)(17)
Therefore, highest prime factor is 17.



u can further break down 145

145 = 29 * 5

Hope it's clear.


That's right. I missed the last option.
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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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New post 30 Oct 2018, 10:27
Bunuel wrote:
What is the largest prime factor of the expression 3^8 − 2^12?

A. 2
B. 3
C. 5
D. 17
E. 29



We see that the expression 3^8 - 2^12 is a difference of squares. Thus, we have:

3^8 - 2^12 = (3^4 + 2^6)(3^4 - 2^6) = (3^4 + 2^6)(3^2 + 2^3)(3^2 - 2^3)

3^4 + 2^6 = 81 + 64 = 145 = 5 x 29

3^2 + 2^3 = 9 + 8 = 17

3^2 - 2^3 = 9 - 8 = 1

So 3^8 - 2^12 = 5 x 29 x 17, and the largest prime factor is 29.

Answer: E
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Re: What is the largest prime factor of the expression 3^8 − 2^12? &nbs [#permalink] 30 Oct 2018, 10:27
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