In the base 10 arithmetic problem , each distinct letter represents a different digit, and leading zeroes are not allowed. What is the maximum possible value of ?
Solution
Clearly , and from the hundreds column, or 9. Since or , it is easy to see that can be at most 7, in which case and must be 8 and 9, so . But because of the tens column, we must have , and in fact since cannot be 0 or , which is impossible given the remaining choices. Therefore, is at most 6. Suppose and . Then we must have and be 7 and 8. With the remaining digits , and 5, we must have in the ones column that and are 2 and 3, which leaves no possibility for . If instead , then and are 7 and 9. Since again and , the only possibility is , and , giving .
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