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

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Math Expert
Joined: 02 Sep 2009
Posts: 58320
What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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10 Aug 2018, 03:24
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Difficulty:

85% (hard)

Question Stats:

44% (01:40) correct 56% (01:52) wrong based on 202 sessions

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

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

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Joined: 25 May 2016
Posts: 85
Location: Singapore
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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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10 Aug 2018, 03:30
2
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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10 Aug 2018, 03:32
2
1
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

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

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10 Aug 2018, 03:33
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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10 Aug 2018, 03:33
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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Posts: 85
Location: Singapore
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Re: What is the largest prime factor of the expression 3^8 − 2^12?  [#permalink]

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10 Aug 2018, 03:59
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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30 Oct 2018, 11:27
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.

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Re: What is the largest prime factor of the expression 3^8 − 2^12?   [#permalink] 30 Oct 2018, 11:27
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