If x is a perfect square, such that 9x^5 has 5 distinct prime factors

Question

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?

(1) x has a pair of consecutive integers as its prime factors
(2) The number of trailing zeroes in x is 0

chetan2u · Answer

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?

Let x =y^2 , where y is an integer. 
 9x^{5}  has 5 distinct prime factors, so x will have 5 distinct prime factors, if it is a multiple of 3, or it will have 4 distinct prime factors.

Now  15\sqrt{x}  will have 
a) 5 distinct prime factors if x is a multiple of 15, 
b) 6 distinct prime factors if x is a multiple of only one of 3 or 5, or 
C) 7 distinct prime factors if x is not a multiple of either of 3 or 5.

1) X has a pair of consecutive integers as its prime factors
So x has 2 and 3 as prime factors. 
But we still do not anything about 5.

2) The number of trailing zeroes in X is 0
Here, we can say nothing about 3 or 5.

Combined
x is an even multiple of 3, but has no trailing zeroes. 
So, x is not a multiple of 5.
Hence, case (b)
Sufficient

sayan640 · Answer

KarishmaB   MartyMurray  Can you please explain in a bit easier way ? with some simple example ?

MartyMurray · Answer

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

If x is a perfect square, such that 9x^5 has 5 distinct prime factors

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