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 x ^{n} – y divisible by p, where p is a prime number?**

**n = p****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, a^{p} – a is divisible by p. Neat, isn’t it?

Lets try, say a = 5 and p = 3

Then a^{p} – a = 5^{3} – 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 x^{p} – y, but still cannot prove divisibility by p

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

x^{p} – (x – sp) = x^{p} – x + sp

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

We go with **option (C)**