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Revision as of 16:19, 17 February 2022
Problem
Find the three-digit positive integer 
whose representation in base nine is 
where 
and 
are (not necessarily distinct) digits.
Solution
We are given that ![\[100a + 10b + c = 81b + 9c + a,\]](http://latex.artofproblemsolving.com/9/8/4/98419c4a18d61bbd8b2c652976c8427a082c8167.png)
which rearranges to ![\[99a = 71b + 8c.\]](http://latex.artofproblemsolving.com/0/2/a/02a0693b866a5a516e33b7d14ea36547a9b2e122.png)
Taking both sides modulo 
we have
The only solution occurs at 
from which 
Therefore, the requested three-digit positive integer is 
~MRENTHUSIASM
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
