HOT Competition 27 Aug/8AM: What is the value of positive integer q?

We can have multiple possibilities here clearly. (p=10, q=15) or (p=20,q=25) and so on. So, (1) alone is not sufficient

Just like with the previous statement, we can have multiple possibilities here (10,17) or (20,27) and so on.

Now, let's consider both the statements together...
Since there is a remainder in either case, we can conclude p and q are not equal. So, either p>q or q>p. Let's consider these cases one after another

Case 1: if p>q
In this case, the remainder of q/p will be equal to q. A number divided by something greater than itself, leaves the entire dividend as remainder. From statement 2, we know that quantity is going to be 7. In other words q = 7.
Now, we have to check if this is even a possibility that satisfies statement (1). q=7 and p=12 satisfies (1). So, yes, this is a possibility

Case 2: if q>p
In this case, the remainder of p/q will be equal to p. From (1) we get p = 5
Now, just like in the previous case, let's see if there's even a possibility that this satisfies the statement (2)
(2) says that any number (q) divided by p(=5) leaves a remainder of 7. Clearly, this is impossible. Becaue 5 can divide 7 at least one more time. In other words, the remainder can never be greater than the divisor. So, case 2 is not possible.

Thus, we have only 1 possibility of q (from case 1) and this is q = 7.
So, (1) & (2) together sufficient to give the value of q. Answer : C