2022 AIME I Problems/Problem 2
Contents
Problem
Find the three-digit positive integer whose representation in base nine is where and are (not necessarily distinct) digits.
Solution 1
We are given that which rearranges to Taking both sides modulo we have The only solution occurs at from which
Therefore, the requested three-digit positive integer is
~MRENTHUSIASM
Solution 2
As shown in Solution 1, we get .
Note that and are large numbers comparatively to , so we hypothesize that and are equal and fills the gap between them. The difference between and is , which is a multiple of . So, if we multiply this by , it will be a multiple of and thus the gap can be filled. Therefore, the only solution is , and the answer is .
~KingRavi
Solution 3
As shown in Solution 1, we get
We list a few multiples of out: Of course, can't be made of just 's. If we use one , we get a remainder of , which can't be made of 's either. So doesn't work. can't be made up of just 's. If we use one , we get a remainder of , which can't be made of 's. If we use two 's, we get a remainder of , which can be made of 's. Therefore we get so and . Plugging this back into the original problem shows that this answer is indeed correct. Therefore,
~Technodoggo
Solution 4
As shown in Solution 1, we get .
We can see that is larger than , and we have an . We can clearly see that is a multiple of , and any larger than would result in being larger than . Therefore, our only solution is . Our answer is .
~Arcticturn
Solution 5
As shown in Solution 1, we get which rearranges to So
vladimir.shelomovskii@gmail.com, vvsss
Solution 6
As shown in Solution 1, we have that .
Note that by the divisibility rule for , we have . Since and are base- digits, we can say that or . The former possibility can be easily eliminated, and thus . Next, we write the equation from Solution 1 as , and dividing this by gives . Taking both sides modulo , we have . Multiplying both sides by gives , which implies . From here, we can find that and , giving an answer of .
~Sedro
Video Solution by OmegaLearn
https://youtu.be/SCGzEOOICr4?t=340
~ pi_is_3.14
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=z5Y4bT5rL-s
Video Solution
https://www.youtube.com/watch?v=CwSkAHR3AcM
~Steven Chen (www.professorchenedu.com)
Video Solution
https://youtu.be/MJ_M-xvwHLk?t=392
~ThePuzzlr
Video Solution by MRENTHUSIASM (English & Chinese)
https://www.youtube.com/watch?v=v4tHtlcD9ww&t=360s&ab_channel=MRENTHUSIASM
~MRENTHUSIASM
Video Solution
~AMC & AIME Training
See Also
2022 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 |
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