2022 AIME I Problems/Problem 2
Contents
Problem
Find the three-digit positive integer whose representation in base nine is where and are (not necessarily distinct) digits.
Solution 1
We are given that which rearranges to Taking both sides modulo we have The only solution occurs at from which
Therefore, the requested three-digit positive integer is
~MRENTHUSIASM
Solution 2
As shown in Solution 1, we get . Note that and are big numbers comparatively to , so we hypothesize that and are equal and fills the gap between them. The difference between and is , which is a multiple of . So, if we multiply this by , it will be a multiple of and thus the gap can be filled. Therefore, a viable solution is and the answer is .
~KingRavi
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=z5Y4bT5rL-s
Video Solution
https://www.youtube.com/watch?v=CwSkAHR3AcM
~Steven Chen (www.professorchenedu.com)
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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