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
.
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