(Russia 2001).
Let a and b be distinct positive integers such that is divisible by
. Prove that
.
Proof.
Let and write
and
with
. Then,
is an integer. N^ote that . Similarly,
.
Because , we have
Now, we apply the following lemma: Let a and b be two coprime numbers. If c is an integer such that , then
.
Hence, we get, implying that
. Therefore,
.
It follows that . QED.
Note that the key step divides g can also be obtained by clever algebraic manipulations such as
.
More Olympiad problems later,
Nalin Pithwa