Difference between revisions of "2020 AIME I Problems/Problem 3"
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== Problem == | == Problem == | ||
| − | A positive integer <math>N</math> has base-eleven representation <math>\underline{a}\underline{b}\underline{c}</math> and base-eight representation <math>\underline1\underline{b}\underline{c}\underline{a},</math> where <math>a,b,</math> and <math>c</math> represent (not necessarily distinct) digits. Find the least such <math>N</math> expressed in base ten. | + | A positive integer <math>N</math> has base-eleven representation <math>\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}</math> and base-eight representation <math>\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},</math> where <math>a,b,</math> and <math>c</math> represent (not necessarily distinct) digits. Find the least such <math>N</math> expressed in base ten. |
== Solution == | == Solution == | ||
Revision as of 21:03, 12 March 2020
Problem
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.
Solution
Since 
, 
, and are digits in base eight, they are all 7 or less. Now, from the given equation, 
. Since , , and have to be positive, 
. If 
, then by casework we see that 
and 
is the only solution. Since the question asks for the lowest working value, we can stop here. Finally, 
, so our answer is 
.
~ JHawk0224
See Also
| 2020 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. 
