2018 AIME I Problems/Problem 2
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 .
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 .
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 It may be verified that
https://www.youtube.com/watch?v=WVtbD8x9fCM ~Shreyas S
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