**(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

### Like this:

Like Loading...

*Related*