2018 AIME I Problems/Problem 2
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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
.
Solutions
Solution Algebra
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
.
-expiLnCalc
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 | ||
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