2013 AIME II Problems/Problem 2
Positive integers and
satisfy the condition
Find the sum of all possible values of
.
Solution
To simplify, we write this logarithmic expression as an exponential one. Just looking at the first log, it has a base of 2 and an argument of the expression in parenthesis. Therefore, we can make 2 the base, 0 the exponent, and the argument the result. That means (because
). Doing this again, we get
. Doing the process one more time, we finally eliminate all of the logs, getting
. Using the property that ${a^{x^{y}}=a^{xy}$ (Error compiling LaTeX. Unknown error_msg), we simplify to
. Eliminating equal bases leaves
. The largest
such that
divides
is
, so we only need to check
,
, and
. When
,
; when
,
; when
,
. Summing all the
's and
's gives the answer of
.