On the test, I will give you the outline of the proof with portions missing, then you will need to fill in those missing phrases. Here is the exact question that will be on the test, with the answers highlighted in red:

Theorem: There are an infinite number of prime numbers.

Proof: Assume _____________  there are a finite number of primes
Then the list of prime numbers would look like: 2, 3, 5, 7, ... , M

Consider the number P, which is constructed from this list so that
P = _____________  2×3×5×7×...×M+1

This leads to a contradiction. P is bigger than the largest prime, M. However, P itself is prime because _____________  it cannot be divided by any other prime number.

This contradiction shows that our assumption was wrong, therefore there are an infinite number of primes.

You will need to be able to fill in those highlighted portions. For a more detailed explanation, see this video.

Infinitude of the Primes Video