1985 AIME Problems/Problem 3
Problem
Find if
,
, and
are positive integers which satisfy
, where
.
Solution
Expanding out both sides of the given equation we have . Two complex numbers are equal if and only if their real parts and imaginary parts are equal, so
and
. Since
are integers, this means
is a divisor of 107, which is a prime number. Thus either
or
. If
,
so
, but
is not divisible by 3, a contradiction. Thus we must have
,
so
and
(since we know
is positive). Thus
.