2020 AIME I Problems/Problem 3
A positive integer has base-eleven representation and base-eight representation where and represent (not necessarily distinct) digits. Find the least such expressed in base ten.
From the given information, . Since , , and have to be positive, . Since we need to minimize the value of , we want to minimize , so we have . Then we know , and we can see the only solution is , . Finally, , so our answer is .
Solution 2 (Official MAA)
The conditions of the problem imply that , so . The maximum digit in base eight is and because , it must be that is or When , it follows that , which implies that . Then must be or If , then is not an integer, and if , then , so . Thus , and . The number also satisfies the conditions of the problem, but is the least such number. Minor edits by TryhardMathlete Video Solution:
|2020 AIME I (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|