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What is greatest prime factor of 4^17-2^28?

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by Brent@GMATPrepNow » Sat Apr 27, 2013 2:52 pm
lukemacdougall wrote:Hey everyone, could someone help explain the solution to the below problem? I know you can simplify 4^17 to 2^34, but don't know where to go from there. thanks!


What is greatest prime factor of 4^17-2^28?

a. 2
b. 3
c. 5
d. 7
e. 11
First of all 4^17 = (2^2)^17 = 2^34

So, 4^17 - 2^28 = 2^34 - 2^28
= 2^28(2^6 - 1) [factored out 2^28]
= 2^28(2^3 + 1)(2^3 - 1) [factored again]
= 2^28(8 + 1)(8 - 1) [evaluated]
= 2^28(9)(7) [evaluated]
= 2^28(3)(3)(7)

So, as you can see, the greatest prime factor is [spoiler]7 (answer = D)[/spoiler]

Cheers,
Brent
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by srcc25anu » Sat Apr 27, 2013 2:54 pm
You are correct. simplify to 2^34 - 2^28
=> 2^28 * (2^6 - 1)
=> 2^28 * (2^3 + 1) * (2^3 + 1)
=> 2^28 * 9 * 7
=> 2^28 * 3^2 * 7

7 is the greatest prime factor.
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by sana.noor » Sun May 05, 2013 9:26 pm
its 7
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