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

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lukemacdougall: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Thu Jan 17, 2013 8:31 pm

by lukemacdougall » Sat Apr 27, 2013 2:15 pm

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

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Source: Beat The GMAT — Problem Solving | Sat Apr 27, 2013 2:15 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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

Brent Hanneson - Creator of GMATPrepNow.com

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srcc25anu: Master | Next Rank: 500 Posts; Posts: 423; Joined: Fri Jun 11, 2010 7:59 amFollowed by: 2 members; Location: Seattle, WA; Thanked: 86 times

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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sana.noor: Legendary Member; Posts: 512; Joined: Mon Jun 18, 2012 11:31 pmFollowed by: 20 members; Thanked: 42 times

by sana.noor » Sun May 05, 2013 9:26 pm

its 7

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