2018 AMC 10A Problems/Problem 25
Problem
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
Solution 1
Observe ; similarly and . The relation rewrites as Since , and we may cancel out a factor of to obtain This is a linear equation in . Thus, if two distinct values of satisfy it, then all values of will. Matching coefficients, we need To maximize , we need to maximize . Since and must be integers, must be a multiple of 3. If then exceeds 9. However, if then and for an answer of . (CantonMathGuy)
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
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All AMC 10 Problems and Solutions |
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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