wied81 wrote:This is from OG 13 #117 in Problem Solving:
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
OA: C
The book posts a pretty obscure way to solve, would like to hear others opinions on this question.
Solution:
It is very unlikely that a problem would require us to calculate 3^8 or 2^8, so we should approach this problem not as an arithmetic question but as an algebraic one.
The first thing we should recognize is that we are being tested on the algebraic factoring technique called the "difference of squares." Recall that the general form of the difference of squares is:
x^2 - y^2 = (x + y)(x - y)
Similarly, we can treat 3^8 - 2^8 as a difference of squares, which can be expressed as:
n = (3^4 + 2^4)(3^4 - 2^4)
We can further factor 3^4 - 2^4 as an additional difference of squares, which can be expressed as:
(3^2 + 2^2)(3^2 - 2^2)
This finally gives us:
n = 3^8 - 2^8 = (3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)
The numbers are now easy to calculate:
n = (81 + 16)(9 + 4)(9 - 4)
n = (97)(13)(5)
We are being asked which of the answer choices is NOT a factor of n, which we have determined to be equal to the product (97)(13)(5). So we must find the answer choice that does not evenly divide into (97)(13)(5).
We immediately see that 97, 13 and 5 are all factors of (97)(13)(5).
This leaves us with 65 and 35. Notice that (97)(13)(5) = (97)(65). Thus, 65 also is a factor of n. Only 35 is not.
Answer:
C
