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

a) There are infinite prime numbers, b) All prime numbers are also abundant numbers

Step-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-B

If n divides A and B, there are integers j, k such that

A = nj and B= nk,

So

A-B= nj - nk = n(j-k)

But j-k is also an integer, which means that n divides also A-B

Now, 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 them

and consider P+1

If 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 pk

Then 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

The problems assigned for the weekend is 60.

A fraction is a non-integer. A fraction usually has a numerator and a denominator. The numerator is the number above. While the denominator is the number below.

Total fraction of assignment done = 3/5 + 1/3

= 14/15

Fraction of assignment left = 1 - 14/15 = 1/15

Number of assignments given = 15 x 4 = 60

To learn more about fractions, please check: link