If x is a perfect square, such that 9x^5 has 5 distinct prime factors, then how many distinct prime factors does 15√x have?
We can ignore the fact that x is perfect square and the fact that we are given information about x and x^5 and then asked about √x since, if x is a perfect square, then x, x^5, and √x have the same prime factors. So, we can treat the question as if x, x^5, and √x are all just x.
According, we can say that the information we are given is that x is an integer and 9x has 5 distinct prime factors.
Now, if 9x has 5 distinct prime factors, x can have 4 or 5 distinct prime factors because 9 has 1 prime factor, 3. So, x may also have 3 among its prime factors and thus have 5 prime factors. At the same time, x could not have 3 among its prime factors and thus have 4 prime factors with the prime factor 3 from 9 being the fifth prime factor of 9x.
We can also say that the question is "How many distinct prime factors does 15x have?"
(1) x has a pair of consecutive integers as its prime factors
The only prime factors that are consecutive integers are 2 and 3. So, this statement tells us that 3 is among the prime factors of x. So, we know from this statement that x has 5 prime factors.
However, we don't know from this statement how many prime factors 15x has. After all, if 5 is a prime factor of x, then 15x has five prime factors. If 5 is not a factor of x, then 15x has 6 prime factors.
Insufficient.
(2) The number of trailing zeroes in x is 0
This choice tells us that x cannot have both 2 and 5 among its prime factors. After all, if 2 and 5 were among x's prime factors, then x would be a multiple of 10 and have at least one trailing 0.
At the same time, since the information from this statement and from the passage does not indicate whether 3 and 5 are factors of x, this statement does not provide sufficient information for determining how many prime factors 15x has.
Insufficient.
(1) and (2) combined
Statement 1 tells us that 2 and 3 are factors of x and that therefore x has 5 factors.
Statement 2 indicates that x cannot have both 2 and 5 among its prime factors. So, since we know from statement 1 that 2 is a prime factor of x, the statements combined indicate that 5 is not a prime factor of x.
So, the statements combined indicate that x has 5 prime factors, one of which is 3 and none of which is 5. Thus, combined, the statements are sufficient for determining that 15x has 6 prime factors.
Sufficient.
Correct answer: C