subirh wrote:

If p and q are two consecutive positive integers and pq= 30x , is x an integer?

(1) p^2 is divisible by 25.

(2) 63 is a factor of q^2

This question is marked as 95% Hard. So, needless to say that this is a Level 750+ question and hence there is no fastest way to solve it. Anyways, following solution is, in one way or other, what everyone above said but I am going to explain it in best way possible.

Universally Available Statements:i)

p and q are two consecutive positive integersi. a) p > = 1; q > = 1

i. b) p - q = 1 (We do not know which one is bigger and it does not matter)

ii)

p*q = 30*x = 2 * 3 * 5 * x.

Question: Is x an Integer?

How to Go about this question: If, Integer x Integer = Integer

How can

x be a Non-Integer?

Possible values of x = 1, 2, 3, \(\frac{1}{2}, \frac{2}{3},\frac{3}{5}, \frac{3}{7}, \frac{7}{30}\) and so on such that given conditions are satisfied.

Solution: If 2, 3 & 5 are factors of p, q, or both, then x would always be an Integer. Hence, the question boils down to determining whether p or q are multiples of 2, 3, and 5.

=> p & q are Consecutive Integers. Therefore, their multiple MUST BE EVEN. Hence, either of p or q is a Multiple of 2.

Now we need to determine whether the given statements are sufficient to determine whether p or q are multiples of 3 & 5.

Statement 1: \(p^2\)is divisible by 25

=> p is a Multiple of 5 -

Insufficient because we don't know whether 3 is a factor of p or q.

Statement 2: 63 is a factor of q^2

=> q = 3*\(\sqrt{7}\)*z (z is some variable which is a factor of x)

Since q is an Integer,

q = 21*z -

Insufficient because we don't know whether 5 is a factor of p or q.

=> q is a Multiple of 3 & 7 -

Insufficient because we don't know whether 5 is a factor of p or q.

Statement 1& 2 together: p is a Multiple of 5 And q is a Multiple of 3 & 7.

=> p*q = Multiple of 2, 3, 5 and 7.

Hence, x will always be an Integer.

Option C - Sufficient.

_________________

I'd appreciate learning about the grammatical errors in my posts

Please hit Kudos If my Solution helps

My Debrief for 750 - https://gmatclub.com/forum/from-720-to-750-one-of-the-most-difficult-pleatues-to-overcome-246420.html

My CR notes - https://gmatclub.com/forum/patterns-in-cr-questions-243450.html