Is ab a prime number?

From Option A:
if b=1 then ab is a prime number else ab is not a prime number .Not sufficient
From Option B:
if b=1 and a is a prime number then ab is a prime number .For other values of a ,ab is not a prime number.
else if b=2,3,4... ab is not a prime number .Not sufficient.

Correct answer is E.

VERITAS PREP OFFICIAL SOLUTION:

Question Type: Yes/No. This question stem asks: “Is ab a prime number?”

Given information in the question stem or diagram: There is no important information given in the question stem.

Statement 1: a is a prime number. This statement is not sufficient but it is tricky. Using conceptual knowledge, many people will assume that the product ab could never be prime. When you multiply any prime number with another positive integer, it could never be prime unless that integer is 1. However, since b could be 1 or any fraction, this statement is not sufficient, and you can eliminate choices A and D. For instance, if a is 7 and b is 1 or 2 7 then the answer is “yes.” However if b is any positive integer other than 1, then the answer is “no.” Note: This question is about avoiding assumptions! Do not assume that b is a whole number, and don’t forget about the very important number 1.

Statement 2: b is not a prime number. If b is not prime then b could easily be a fraction or equal to 1. In either case it is very possible for ab to be a prime number and it is very easy to show that it might not be. Not sufficient. Eliminate answer B.

Together: Taken together you know that a is prime and b is not. But this leaves you in the same dilemma as with Statement 1 alone. Certainly a and b can multiply to equal a non-prime number such as “a = 2 and b = 6.” 12 is not a prime number. However, if b =1 then whatever prime number a equals will remain and ab will be that prime number. As in the first statement, the key in this problem is to avoid assumptions and remember that nothing limits b to an integer and nothing excludes the value of 1. Answer is E.

Statement 1. A is a prime number.
Case 1. If ‘b’ = 1; ab is a prime number.
Case 2. If ‘b’ not equal to 1; ab is not a prime number. Hence, Insufficient.
Statement 2. ‘b’ is not a prime number.
Case 1. Let b = 4(non-prime) and a = 1/2. Then, ab = 2 i.e. a prime number.
Case 2. Let b = 4(non-prime) and a = 1. Then ab = 4 i.e. a composite number.Hence, Insufficient.
Statement 1 & 2 together. ‘a’ is a prime and ‘b’ is a non- prime number.
Case 1. Let a = 2(Prime number) and b = 1 (non-prime number). Then, ab = 2 i.e. a prime number.
Case 2. Let a = 2(Prime number) and b = 4 (non- prime number). Then, ab= 8 i.e. non-prime number. Hence, Insufficient.
Hence, e is the answer.

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