EgmatQuantExpert wrote:
Question:
How many prime factors does the number X have?
1) X is divisible by 53
2) X is divisible by 4 distinct integers.
A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient
Statement 1If X is divisible by 53, then X could just be '53' (in which case only 1 prime factor, 53) or X could be '53*2 = 106' (in which case 2 prime factors, 2 and 53).
So we cannot be sure about the number of prime factors of X.
Not Sufficient.
Statement 2X is divisible by 4 distinct integers, this means X has total 4 factors. Now any number having 4 factors in total, could have two kinds of prime factorisation:
a) X = p^3. So X could be cube of a prime number. In this case X will have total 4 factors, but only 1 prime factor (p).
b) X = p1 * p2. So X could be a product of two distinct prime numbers. In this case also X will have total 4 factors, but there are 2 prime factors (p1 and p2).
So we cannot be sure about number of prime factors of X.
Not Sufficient.
Combining the statements, we know X is divisible by 53 and X has 4 total factors.
So if X looks like p^3, then X = 53^3. In this case X has only 1 prime factor 53.
But if X looks like p1 * p2, then X = 53*p2 (product of 53 with any other prime number). In this case X has 2 prime factors 53 and p2.
So we still cannot be sure, as there are two possibilities.
Not Sufficient.
Hence
E answer