Problem:
Let the letters , , , , , represent distinct digits. Suppose is the greatest number that satisfies the equation
What is the value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Firstly, note that and, similarly, , so the equation can be simplified to .
For to remain three digits, must be . Moreover, must also be less than to avoid carrying over to the hundreds digit and making the product digits. Since we need to be the greatest number, must be . To identify the possible values for , we note that so far we have , so we must avoid carrying to the tens digit to keep the resulting product three digits. Hence, .
We can try and verify that the resulting product has unique digits that haven't been used yet: , which does not have unique digits. Trying , we get , which satisfies our criteria. Hence,
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions