Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2020 AIME I Problems/Problem 3"

(→‎Solution)
(→‎Video Solution)
 
(12 intermediate revisions by 9 users not shown)
Line 3: Line 3:
 
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 ==
+
== Solution 1 ==
 
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>.
  
 
~ JHawk0224
 
~ JHawk0224
  
−
Video Solution:
+
==Solution 2 (Official MAA)==
 +
The conditions of the problem imply that <math>121a + 11b + c = 512 + 64b + 8 c + a</math>, so <math>120 a = 512+ 53b+7c</math>. The maximum digit in base eight is <math>7,</math> and because <math>120a \ge 512</math>, it must be that <math>a</math> is <math>5, 6,</math> or <math>7.</math> When <math>a = 5</math>, it follows that <math>600=512 + 53b+7c</math>, which implies that <math>88 = 53b+7c</math>. Then <math>b</math> must be <math>0</math> or <math>1.</math> If <math>b = 0</math>, then <math>c</math> is not an integer, and if <math>b = 1</math>, then <math>7c = 35</math>, so <math>c = 5</math>. Thus <math>N = 515_{11}</math>, and <math>N=5\cdot 121 + 1\cdot 11 + 5 = 621</math>. The number <math>637_{11} =1376_{8} = 766</math> also satisfies the conditions of the problem, but <math>621</math> is the least such number.
 +
 
 +
==Video Solution==
  
 
https://youtu.be/hZSBUXCX5hI
 
https://youtu.be/hZSBUXCX5hI
 +
 +
Minor edits by TryhardMathlete
 +
 +
== Video Solution by OmegaLearn==
 +
https://youtu.be/mgEZOXgIZXs?t=1204
 +
 +
~ pi_is_3.14
 +
 +
==Video Solution==
 +
https://youtu.be/SuVsBIz8pZ8
  
 
==See Also==
 
==See Also==

Latest revision as of 03:48, 21 January 2023

Problem

A positive integer $N$ has base-eleven representation $\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}$ and base-eight representation $\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.

Solution 1

From the given information, $121a+11b+c=512+64b+8c+a \implies 120a=512+53b+7c$. Since $a$, $b$, and $c$ have to be positive, $a \geq 5$. Since we need to minimize the value of $n$, we want to minimize $a$, so we have $a = 5$. Then we know $88=53b+7c$, and we can see the only solution is $b=1$, $c=5$. Finally, $515_{11} = 621_{10}$, so our answer is $\boxed{621}$.

~ JHawk0224

Solution 2 (Official MAA)

The conditions of the problem imply that $121a + 11b + c = 512 + 64b + 8 c + a$, so $120 a = 512+ 53b+7c$. The maximum digit in base eight is $7,$ and because $120a \ge 512$, it must be that $a$ is $5, 6,$ or $7.$ When $a = 5$, it follows that $600=512 + 53b+7c$, which implies that $88 = 53b+7c$. Then $b$ must be $0$ or $1.$ If $b = 0$, then $c$ is not an integer, and if $b = 1$, then $7c = 35$, so $c = 5$. Thus $N = 515_{11}$, and $N=5\cdot 121 + 1\cdot 11 + 5 = 621$. The number $637_{11} =1376_{8} = 766$ also satisfies the conditions of the problem, but $621$ is the least such number.

Video Solution

https://youtu.be/hZSBUXCX5hI

Minor edits by TryhardMathlete

Video Solution by OmegaLearn

https://youtu.be/mgEZOXgIZXs?t=1204

~ pi_is_3.14

Video Solution

https://youtu.be/SuVsBIz8pZ8

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_AIME_I_Problems/Problem_3&oldid=187216"