Difference between revisions of "2022 AIME I Problems/Problem 2"
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| − | /* | + | == Problem == |
| + | |||
| + | Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits. | ||
| + | |||
| + | == Solution == | ||
| + | |||
| + | We are given that <cmath>100a + 10b + c = 81b + 9c + a,</cmath> which rearranges to <cmath>99a = 71b + 8c.</cmath> | ||
| + | Taking both sides modulo <math>71,</math> we have | ||
| + | <cmath>\begin{align*} | ||
| + | 28a &\equiv 8c \pmod{71} \\ | ||
| + | 7a &\equiv 2c \pmod971}. | ||
| + | \end{align*}</cmath> | ||
| + | The only solution occurs at <math>(a,c)=(2,7),</math> from which <math>b=2.</math> | ||
| + | |||
| + | Therefore, the requested three-digit positive integer is <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.</math> | ||
| + | |||
| + | ~MRENTHUSIASM | ||
| + | |||
| + | ==See Also== | ||
| + | {{AIME box|year=2022|n=I|num-b=2|num-a=4}} | ||
| + | {{MAA Notice}} | ||
Revision as of 17:18, 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
\begin{align*}
28a &\equiv 8c \pmod{71} \\
7a &\equiv 2c \pmod971}.
\end{align*} (Error compiling LaTeX. Unknown error_msg)
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 2 |
Followed by Problem 4 | |
| 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. 
