&, #, and each represents a different digit, and & multiplied by #

VERITAS PREP OFFICIAL SOLUTION:


At first glance, this problem looks like nothing you’ve seen before! But you do know some things here. When multiplying two-digit numbers, the first step is to multiply the units digits. And here, # * & provides a units digit of #. How is that true, particularly if as stated above the full product is less than ten? That means that & must equal 1, so that when it multiplies with another number, that other number (#) stays the same. 1 * # = #, so we can prove that & is 1.

We also know that the digits must be unique, so choice A is eliminated, and the fact that the & is 1 means that D and E are eliminated (the first digit must be a 1). So that leaves just two options, 12 and 13. And we know that the problem, then, is either 12*21 or 13*31, and we have to have a solution with the units and hundreds digits the same. At this point, we have two quick multiplication problems to do:

12*21 = 252 (this works)

13*31 = 403 (this does not)

So the correct answer must be 12. The key is to begin with a first step, taking inventory and advantage of what you know and working from there. Difficult problem solving questions often hinge on the exam’s knowledge that people are uncomfortable when they cannot see the entire process all at once, but remember that they also have to be written such that many examinees can solve them in 2 minutes. So rest assured that even if you do not know much, you will still know enough to get started. Great test takers always begin by taking what they know and working from there; those who don’t score as well tend to focus more on what they don’t know, and struggle to get started.