GMAT Math : Fermat’s little theorem

Fermat’s little theorem is a nifty thing to remember when dealing with prime divisibility questions in the GMAT Quant section

First let’s start with an example:

Is xn – y divisible by p, where p is a prime number?

  1. n = p
  2. y = x – sp where s is an integer

Before we get muddled with putting random values and testing for the statements. Here’s a neat trick/theory by the name of Fermat’s little theorem

For any integer a, ap – a is divisible by p. Neat, isn’t it?

Lets try, say a = 5 and p = 3

Then ap – a = 53 – 5 = 120 which is divisible by 3!

Coming to the original question,

With statement (1), we can change the form of the original equation to xp – y, but still cannot prove divisibility by p

With the additional statement (2), we can change the form of the original equation to

xp – (x – sp) = xp – x + sp

Now, we know (xp – x) is divisible by p (Thanks to Fermat!) and sp is divisible by p since s is an integer so xp – x + sp is divisible by p

We go with option (C)