Difference between revisions of "2018 AIME I Problems/Problem 2"
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We can have <math>a=4,8,12</math>, but only <math>a=4</math> falls within the possible digits of base <math>6</math>. Thus <math>a=4</math>, <math>c=1</math>, and thus you can find <math>b</math> which equals <math>10</math>. Thus, our answer is <math>4\cdot225+1\cdot15+10=\boxed{925}</math>. | We can have <math>a=4,8,12</math>, but only <math>a=4</math> falls within the possible digits of base <math>6</math>. Thus <math>a=4</math>, <math>c=1</math>, and thus you can find <math>b</math> which equals <math>10</math>. Thus, our answer is <math>4\cdot225+1\cdot15+10=\boxed{925}</math>. | ||
+ | ==Solution 3 (Official MAA)== | ||
+ | The problem is equivalent to finding a solution to the system of Diophantine equations <math>196a+14b+c=225a+15c+b</math> and <math>225a+15c+b=216a+36c+6a+c,</math> where <math>1\le a\le 5,\,0\le b\le 13,</math> and <math>0\le c\le 5.</math> Simplifying the second equation gives <math>b=22c-3a.</math> a. Substituting for b in the first equation and simplifying then gives <math>a=4c,</math> so <math>a = 4</math> and <math>c = 1,</math> and the base-<math>10</math> representation of <math>n</math> is <math>222 \cdot� 4 + 37 \cdot� 1 = 925.</math> It may be verified that <math>b=10\le 13.</math> | ||
==Video Solution== | ==Video Solution== | ||
Revision as of 20:37, 26 February 2021
Contents
Problem
The number can be written in base as , can be written in base as , and can be written in base as , where . Find the base- representation of .
Solution 1
We have these equations: . Taking the last two we get . Because otherwise , and , .
Then we know . Taking the first two equations we see that . Combining the two gives . Then we see that .
Solution 2
We know that . Combining the first and third equations give that , or The second and third gives , or We can have , but only falls within the possible digits of base . Thus , , and thus you can find which equals . Thus, our answer is .
Solution 3 (Official MAA)
The problem is equivalent to finding a solution to the system of Diophantine equations and where and Simplifying the second equation gives a. Substituting for b in the first equation and simplifying then gives so and and the base- representation of is $222 \cdot� 4 + 37 \cdot� 1 = 925.$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character � (U+FFFD)) It may be verified that
Video Solution
https://www.youtube.com/watch?v=WVtbD8x9fCM ~Shreyas S
See Also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.