The Quest for 700: Weekly GMAT Challenge (Answer)

Write A as 10x + y, where x is the tens digit and y is the units digit. We know that B is a smaller two-digit integer, and that B is smaller by a particular amount: twice A’s units digit, or 2y.

So B is 10x + y – 2y = 10x – y.

We can think about a few constraints. Since B is “smaller,” we know that y cannot be 0 (otherwise, B wouldn’t be smaller than A). Likewise, we know that x is not 0 (otherwise, A would not be a two-digit integer). Finally, we know that x is not 1. Otherwise, B would not be a two-digit integer either, since B = 10x – y. If x were equal to 1, then B would equal 10 – a positive digit, which would be a single-digit number.

We want to know the hundreds digit of the product of A and B. Write this product in terms of x and y:

AB = (10x + y)(10x – y) = 100x2 – y2.

Notice that we get the difference of squares. Also, since y is a single positive digit, the greatest y2 can be is 81, while the smallest is 1. Meanwhile, x2 is multiplied by 100. For instance, if x = 4, then x2 = 16, and the first term above is 1,600. Then we subtract a number between 1 and 81 (inclusive), so we get 1,599 down to 1,519. In either case, we have a specific hundreds digit (5) that doesn’t depend on y, the units digit, at all.

In fact, AB’s hundreds digit is completely dictated by the units digit of x2.

So the question can be rephrased: what is the units digit of x2, where x is the tens digit of A?

Statement (1): NOT SUFFICIENT. We know that x could be 2, 3, 5, or 7. These are the only prime digits. Squaring those digits, we get 4, 9, 25, and 49. The units digits are different, so this statement is not sufficient.

Statement (2): NOT SUFFICIENT. We know that x could be 3, 4, 6, 7, 8, or 9 (since 10 is divisible by 2 and 5). Squaring a few of these digits, we get 9, 16 – stop. The units digits are again different, so the statement is not sufficient.

Statements (1) and (2) together: SUFFICIENT. Putting the statements together, we know that x could be 3 or 7. The squares are 9 and 49. Since the units digits are the same, we have sufficiency.