Explanation

This question describes a number q that has digits something like 1p, or 2p, and so on. The thing that the question really wants from us is the main unknown that it hasn't assigned a variable to, which is the tens digit of the number. Let's turn to the data statements, separately first.

Statement (1) describes a relationship between the tens digit and the units digit. We can think of cases. The number q could be 16, since . The complete set of possibilities is 16, 27, 38, or 49. Since those are the only four possibilities, and they all lie between 10 and 50, this statement gives us sufficient information to answer the question definitively. Statement (1) is sufficient.

Statement (2) gives us a relationship between the units digit and the number as a whole. Again, we'll explore cases. Let's say . Then q would be 3, but that's not allowed, since q is a two-digit number. So p can't be 1. Similarly, yields 7, which is not two digits. Finally, gives us a two-digit number, 11, but that's also not allowed, because 3 is supposed to be the units digit of q, and 3 isn't 1. Proceeding, we find 4 gives 15, out; 5 gives 19, out; 6 gives 23, out; 7 gives 27, in!; 8 gives 31, out; and 9 gives 35, out. There is only one two-digit number q with a units digit p that satisfies , and that is 27. Since we have defined q, we have more than enough information to answer the question definitively. Statement (2) is sufficient.

The correct answer is (D).