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.
p is a multiple of 5.
so far so good ..
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