Problem Solving

Vincen wrote: ↑

Thu Nov 05, 2020 4:32 am

If \(n = 3^8 - 2^8,\) which of the following is NOT a factor of \(n?\)

(A) 97
(B) 65
(C) 35
(D) 13
(E) 5

Answer: C

Source: Official Guide

This is really an algebra question in disguise.
Recognize that n = 3^8 - 2^8 is a difference of squares, and we can factor these.
For example, x^2 - y^2 = (x + y)(x - y)
Similarly, x^10 - y^10 = (x^5 + y^5)(x^5 - y^5)

Now onto the question.

n = 3^8 - 2^8
= (3^4 + 2^4)(3^4 - 2^4)
= (3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)
= (3^4 + 2^4)(3^2 + 2^2)(3 + 2)(3 - 2)
= (97)(13)(5)(1)

We can see that 97 is a factor of n. ELIMINATE A
We can see that 65 is a factor of n (since 13x5 = 65). ELIMINATE B
We can see that 13 is a factor of n. ELIMINATE D
We can see that 5 is a factor of n. ELIMINATE E

Answer: C

Cheers,
Brent