If z = x*y^3, where x and y are positive integers, what is the value

Given

\( z = x * y^3\)

\(x \geq{1}\) ; \(y \geq{1}\)

Question

y = ?

Statement 1

The number of different positive factors of z is 4.

\( z = x * y^3\)

Now y can be = 1, and x can be a cube of a prime number.

So \(x = p^3\)

\(z = p^3 * 1\)

Alternatively,

y can be any prime number and x = 1

Hence, we can eliminate A and D

Statement 2

z is the product of two different prime numbers.

So, \(z = p_1 * p_2\)

Also \( z = x * y^3\)

Now there can be multiple scenarios we can think of

1. y is a prime number and x is also a prime number : In that case, z is a multiple of more than two prime numbers, hence this is not a valid case.
   
2. y is a prime number and x = 1 : In that case, z is again a multiple of more than two prime numbers, hence this is also not a valid case.
   
3. \(x = p_1 * p_2\) and y = 1 : This is valid.

Note: We don't even have to consider scenario 1 and scenario 2 (to save time). It's given that y is an integer, so if y has any prime factors in it, z will be a product of more than two prime factors. Hence y has to be equal to 1.

Thus have a definite answer.

IMO B