2002 Indonesia MO Problems/Problem 1
Problem
Show that is divisible by
for any integers
.
Solution
In order for to be divisible by
, it must be divisible by
and
. Note that
can be factored into
.
If is even, then
. If
, then
, and if
, then
. That means for all positive
,
is divisible by
.
If , then
. If
, then
. If
, then
. That means for all positive
,
is divisible by
.
Because is divisible by
and
, it must be divisible by
.