(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

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1

no

step-by-step explanation: