Solving the Problem

You might assume that you can plug $725$,$278 \times 67$,$066$ into the crappy GRE calculator, but you'll quickly see that you get a nice ERROR message.

There is good news, however. To find the tens digit and unit digit of the product, you don't need to multiply the two numbers together in their entirety. It's enough to take just the last two digits of each number:

$78 \times 66 = 5$,$148$

Now we can add the tens digit and unit digit together: $4+8=12$. Thus, the correct answer is A.