Difference between revisions of "2020 AIME I Problems/Problem 3"
(→Solution) |
|||
Line 2: | Line 2: | ||
== 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. | ||
== Solution == | == Solution == |
Revision as of 17:06, 12 March 2020
Note: Please do not post problems here until after the AIME.
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.