Difference between revisions of "2018 AIME I Problems/Problem 2"
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Then we know <math>3a+b=22</math>. | Then we know <math>3a+b=22</math>. | ||
− | Taking the first two equations we see that <math>29a+14c=13b</math>. Combining the two gives <math>a=4, c= | + | Taking the first two equations we see that <math>29a+14c=13b</math>. Combining the two gives <math>a=4, b=10, c=1</math>. Then we see that <math>222 \times 4+37 \times1=\boxed{925}</math>. |
==Solution 2== | ==Solution 2== |
Revision as of 19:22, 31 December 2020
Contents
[hide]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
.
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 |
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