sanju09 wrote:If p is a prime number, which of the following CANNOT be a prime number?
A. p + 1
B. p + 2
C. 3p + 2
D. 5p + 3
E. p² - 16

Good question, Sanjeev.

Here's one approach:

We're looking for an expression that CANNOT be a prime number. So, we can ELIMINATE any expression that CAN be a prime number.

A. p + 1 If p = 2, then p + 1 = 3, which is prime. ELIMINATE A
B. p + 2 If p = 3, then p + 2 = 5, which is prime. ELIMINATE B
C. 3p + 2 If p = 3, then 3p + 2 = 11, which is prime. ELIMINATE C
D. 5p + 3 If p = 2, then 5p + 3 = 13, which is prime. ELIMINATE D

By the process of elimination, we can be certain that the correct answer must be E

NOTE: We need not confirm that answer choice E is the correct answer (i.e., confirm that p² - 16 cannot be a prime number). In fact, doing so would be a waste of time on test day, when time plays a huge role.

Cheers,
Brent

sanju09 wrote:If p is a prime number, which of the following cannot be a prime number?
A. p + 1
B. p + 2
C. 3p + 2
D. 5p + 3
E. p² - 16

Here's an algebraic approach.

Answer choice E: p² - 16 = (p + 4)(p - 4)
Since p² - 16 can be written as the product of two different integers, we might automatically conclude that p² - 16 CANNOT be prime. However, if a number can be written as the product of two different integers, it's still possible for that number to be prime, if one of the two numbers in the product is 1. For example, we can write the prime number 7 as the product (7)(1).
So, before we draw the conclusion that p² - 16 cannot be prime, we must first rule out the possibility that either (p + 4) or (p - 4) equals 1.
So, let's rule out this possibility.
What about (p + 4)? Since we're told that p is prime, we know that p is POSITIVE, which means (p + 4) cannot equal 1.
What about (p - 4)? Well, (p - 4) can equal 1 when p = 5.
When we plug p = 5 into the expression p² - 16, we get 9, and 9 is not prime.

So, we can conclude with certainty that p² - 16 can never be prime.
Answer: E

Cheers,
Brent

sanju09 wrote:
Thanks for the like, Brent. Your explanation is to the point. How about this?
https://www.beatthegmat.com/is-4t-3-t-3- ... tml#716881

I've responded to that question.

Cheers,
Brent