What is the greatest prime factor of 6^8 - 3^8 ?

[B BTGModeratorVI]

What is the greatest prime factor of 6^8 − 3^8 ?

A) 3
B) 11
C) 17
D) 19
E) 31

Answer: C
Source: Veritas Prep

[Brent@GMATPrepNow]

What is the greatest prime factor of 6^8 − 3^8 ?

A) 3
B) 11
C) 17
D) 19
E) 31

Answer: C
Source: Veritas Prep

6^8 − 3^8 is a DIFFERENCE OF SQUARES. So we can factor it.
6^8 − 3^8 = (6^4 + 3^4)(6^4 - 3^4)
= (6^4 + 3^4)(6^2 + 3^2)(6^2 - 3^2)
= (6^4 + 3^4)(6^2 + 3^2)(6 + 3)(6 - 3)
= (6^4 + 3^4)(45)(9)(3)
= (6^4 + 3^4)(3)(3)(5)(3)(3)(3)

Hmmmm, we can see that the correct answer is "hiding" in the first number (6^4 + 3^4)
Let's factor out the 3^4, to get:
6^4 + 3^4 = 3^4(2^4 + 1)
= 3^4(16 + 1)
= 3^4(17)
= (3)(3)(3)(3)(17)

So, 6^8 − 3^8 = (3)(3)(3)(3)(17)(3)(3)(5)(3)(3)(3)

So the correct answer is C

ASIDE: For more on factoring differences of squares, see our free video - https://www.gmatprepnow.com/module/gmat- ... /video/955

Cheers,
Brent

[Scott@TargetTestPrep]

What is the greatest prime factor of 6^8 − 3^8 ?

A) 3
B) 11
C) 17
D) 19
E) 31

Answer: C

Solution:

Let’s prime factorize the given expression:

6^8 - 3^8

3^8(2^8 - 1)

3^8(2^4 - 1)(2^4 + 1)

3^8(16 - 1)(16 + 1)

3^8(15)(17)

3^8 x 3 x 5 x 17

So the largest prime factor is 17.

Answer: C