2．A student is attempting to prove that there are infinitely many prime numbers．
The student＇s attempt to prove this is in the box below． Assume there are only finitely many prime numbers，then there is a biggest prime number，\(p\) ． Let \(n = 2 p + 1\) ．Then \(n\) is bigger than \(p\) and since \(2 p + 1\) is not divisible by \(p\) ， \(n\) is a prime number． Hence \(n\) is a prime number bigger than \(p\) ，contradicting the initial assumption． So we conclude there are infinitely many prime numbers．
（a）Use \(p = 7\) to show that the following claim made in the student＇s proof is not true： since \(2 p + 1\) is not divisible by \(p , n\) is a prime number． The student changes their proof to use \(n = 6 p + 1\) instead of \(n = 2 p + 1\)
（b）Show，by counter example，that this does not correct the student＇s proof．
（c）Write out a correct proof by contradiction to show that there are infinitely many prime numbers．