ABC
+ BCB
CDD
In the addition shown above, A, B, C, and D represent the nonzero digits of three 3-digit numbers. What is the largest possible value of the product of A and B ?
8
10
12
14
18
Explanation: First, note that A, B, C, and D have to be digits between 1 and 9 (and the problem does not prevent some letters from having the same value).
The two rightmost columns both contain C + B = D. From that information, you can deduce that B + C < 9. (If B + C were 10 or more, then the rightmost column would “carry over” into the next column, making the tens digit into D + 1 rather than D.)
Next, A + B = C. A larger value of C, then, will reduce B (because B + C < 9) and therefore increase A (because A + B = C).
Since the questions asks to maximize the product for A and B, consider only the cases in which B + C = 9. Further, since A + B = C, you also know that B must be less than C. (Not sure why? Test a couple of real numbers to figure it out.) That leaves only a few cases, so write them out:
If B = 1 and C = 8, then A = 7. In this case, the product of A and B is 7.
If B = 2 and C = 7, then A = 5. In this case, the product of A and B is 10.
If B = 3 and C = 6, then A = 3. In this case, the product of A and B is 9.
If B = 4 and C = 5, then A = 1. In this case, the product of A and B is 4.
That’s all of the possible cases. The largest possible product is 10.
The correct answer is B.
How do you know to make the intuitive leap in the bold portion, to know that you should focus only on cases where B+C=9?