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 (Simple)
As in the previous solution, we get . 99 and 71 are big numbers comparatively to 8, so we hypothesize that
and
are equal and
fills the gap between them. The difference between 99 and 71 is 28, which is a multiple of 4, so if we multiply this by 2, it will be a multiple of 8 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 | ||
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