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

The first thing to consider is whether this can be done by brute force. Personally, I am a fan of math facts, and I always recommend that my students know squares, square roots, and powers of primes – especially 2 because most exponential growth problems involve powers of two. You can check your own knowledge of the facts I recommend with Math FACTor 1. After working on your math facts, the world of numbers will break down into, friends, acquaintances, and strangers. Unfortunately, \(3^8\) is a stranger.

Now, it’s important to mention that most strangers are not so fearsome. In fact, \(3^8 = (3^4)(3^4) = 81(81) = (80 + 1)(80 + 1)\), and if you know your perfect square formula like the back of your hand, as you most definitely should, you can quickly calculate \((80 + 1)(80 + 1) = 6400 + 2(80)(1) + 1 = 6561\). So, the question becomes do we want to prime factor \(6561 – 256\). and if we’re not feeling up to that, do we want to check for divisibility by 97, 65, etc. I’m thinking, “no thanks” on both counts.

Brute force is out. Now what? More algebra with numbers: \(3^8\) and \(2^8\) are both perfect squares!!! Do you remember your perfect squares formula:

\(x^2 – y^2 = (x + y)(x – y)\)

So, \(n = 3^8 – 2^8 = (3^4 – 2^4)(3^4 + 2^4) = (81 – 16)(81 + 16) = 97(65)\). Thus \(n\) is divisible by 97, 65, 13, and 5, but NOT 35.