Let a and b be distinct positive integers such that is divisible by . Prove that
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,