(a) How many prime numbers are (b) How many prime numbers are also abundant numbers?

Answer:a) There are infinite prime numbers, b) All prime numbers are also abundant numbersStep-by-step explanation:To prove a) let's first prove that if n divides both integers A and B then also divides the difference A-BIf n divides A and B, there are integers j, k such thatA = nj and B= nk, SoA-B= nj - nk = n(j-k)But j-k is also an integer, which means that n divides also A-BNow, to prove that there are infinite prime numbers , we will proceed with Reductio ad absurdum.We will suppose that there are only a finite number of primes and then arrive to a contradiction.Suppose there are only n prime numbers,{p1,p2,... pn}then take P=p1.p2...pn the product of all of themand consider P+1If P+1 is prime the proof is complete for P+1 is not in the list.if P+1 is not prime then by the Fundamental Theorem of Arithmetic there is a prime in the list that must divide P+1, let's say pkThen pk also divides P+1-P=1 which is a contradiction because no prime divides 1.b) To prove this, recall that an abundant number is a number for which the sum of its proper divisors is greater than the number itself. Given that a prime number P is only divided by P and 1, the sum of its divisors is P+1 which is greater than P. So P is abundant