Data Sufficiency

AAPL wrote:Manhattan Prep

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.

OA C

Since p and q are two consecutive positive integers, we have q = p + 1.

We have to determine whether pq = p(p + 1) = 30x.

Let's take each statement one by one.

1) p^2 is divisible by 25.

=> p is divisible by 5. Say p = 5m; thus, q = 5m + 1; m is a positive integer

From p(p + 1) = 30x, we have 5m*(5m + 1) = 30x => m(5m +1 ) = 6x. If m = 1, we have x = 1, an integer, the answer is yes; however,
if say m = 2, we have x = 11/3, not an integer, the answer is no. No unique answer. Insufficient.

2) 63 is a factor of q^2.

=> 3^2*7 is a factor of q^2.
=> q is divisible by 3*7 = 21; given that q is an integer.

Say q = 21n; thus, p = 21n - 1; ; m is a positive integer

From p(p + 1) = 30x, we have (21n - 1)(21n) = 30x. If n = 1, we have x = 14, an integer, the answer is yes; however,
if say n = 2, we have x ≠ an integer, the answer is no. No unique answer. Insufficient.

(1) and (2) together

From both the statements, we have 5m = 21n - 1

=> m = (21n - 1)/5 = (20n + n - 1)/5 = 4n + (n - 1)/5

Since m and n are positive integers, the minimum value of n must be 6.

Thus, p = 21n - 1 = 21*6 - 1 = 125 => q = 126

Thus, pq = 125*126 = (5^3)*(6*21) = 30*(25*21) = 30x => x = 25*21, an integer. Sufficient.

The correct answer: C

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: GRE Manhattan | ACT Prep Courses San Francisco | IELTS Prep Courses Boston | Seattle IELTS Tutoring | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.