Difference between revisions of "2020 AIME I Problems/Problem 3"
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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. | 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 1 == | |
| − | |||
| − | |||
| − | == Solution == | ||
From the given information, <math>121a+11b+c=512+64b+8c+a \implies 120a=512+53b+7c</math>. Since <math>a</math>, <math>b</math>, and <math>c</math> have to be positive, <math>a \geq 5</math>. Since we need to minimize the value of <math>n</math>, we want to minimize <math>a</math>, so we have <math>a = 5</math>. Then we know <math>88=53b+7c</math>, and we can see the only solution is <math>b=1</math>, <math>c=5</math>. Finally, <math>515_{11} = 621_{10}</math>, so our answer is <math>\boxed{621}</math>. | From the given information, <math>121a+11b+c=512+64b+8c+a \implies 120a=512+53b+7c</math>. Since <math>a</math>, <math>b</math>, and <math>c</math> have to be positive, <math>a \geq 5</math>. Since we need to minimize the value of <math>n</math>, we want to minimize <math>a</math>, so we have <math>a = 5</math>. Then we know <math>88=53b+7c</math>, and we can see the only solution is <math>b=1</math>, <math>c=5</math>. Finally, <math>515_{11} = 621_{10}</math>, so our answer is <math>\boxed{621}</math>. | ||
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~ pi_is_3.14 | ~ pi_is_3.14 | ||
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| + | ==Video Solution== | ||
| + | https://youtu.be/SuVsBIz8pZ8 | ||
==See Also== | ==See Also== | ||
Revision as of 01:26, 16 February 2021
Contents
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 1
From the given information, 
. Since 
, 
, and have to be positive, 
. Since we need to minimize the value of 
, we want to minimize , so we have 
. Then we know 
, and we can see the only solution is 
, 
. Finally, 
, so our answer is 
.
~ JHawk0224
Solution 2 (Official MAA)
The conditions of the problem imply that 
, so 
. The maximum digit in base eight is 
and because 
, it must be that is 
or 
When , it follows that 
, which implies that 
. Then must be 
or 
If 
, then is not an integer, and if 
, then 
, so 
. Thus 
, and 
. The number 
also satisfies the conditions of the problem, but 
is the least such number.
Video Solution:
Minor edits by TryhardMathlete
Video Solution
https://youtu.be/mgEZOXgIZXs?t=1204
~ pi_is_3.14
Video Solution
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. 
