souvik101990 wrote:
Find the number of factors of the number p, if p is a positive integer less than 100.
1. The number p has odd number of factors.
2. The number p can be expressed as \(x^{2}\), \(y^{3}\) or \(z^{6}\) where x, y, and z are positive 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.
Answer is Option E.
Statement 1: There are many number <=100 which have odd factors. Examples: 2^2, 2^4, 2^6, etc..In these examples the factors will be 3, 5, and 7 respectively. Hence, this can't give us an answer. Thus,
INSUFFICIENT.Statement 2: Lets take number 64 because if we take any other number raised to power 6 it will be >100. 64 can be written as 2^6, 4^3, and 8^2. Even in this case we have different number of factors i.e. 7, 4, and 3. Thus also
INSUFFICIENT.
Taking Statement 1 and 2 together:
Statement 1 says p must have odd factors, hence we can only take 2^6 and 8^2. But even among these 2 we have different number of factors, i.e. 7 and 3. So, even after combining 2 statements we cant solve it.
Hence E