# Understand the problem

Let and be positive integers greater than , with being a prime. Show that if divides and divides , then is a perfect square.

##### Source of the problem

Argentina TST 2005

##### Topic

Number Theory

##### Difficulty Level

Easy

##### Suggested Book

An Excursion in Mathematics

# Start with hints

Do you really need a hint? Try it first!

Note that, divides either or . Using inequalities, show that has to be greater than . Hence .

Write for some integer . Using the fact that , show that the integer has residue 1 modulo .

Note that, if , then has to be greater than or equal to . Show that this leads to a contradiction.

As has residue 1 modulo , we have $p\ge n+1$. This means that . This is clearly a contradiction. Hence . Therefore .

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