# Understand the problem

Suppose that for a prime number and integers the following holds:

Prove that .

Prove that .

##### Source of the problem

Baltic Way 2009

##### Topic

Number Theory

##### Difficulty Level

Medium

##### Suggested Book

Elementary Number Theory by David Burton

# Start with hints

Do you really need a hint? Try it first!

Show that, modulo . Hence . From there, try to find a relation between and modulo .

As , we have . Note that, is an integer and hence . Using Fermat’s little theorem, conclude that . Use this along with to conclude that either or are both divisible by .

Clearly, if then we are done. Prove that the case can be reduced to this one.

If , then we have and . These together give . Using , conclude that . The proof is now concluded using hint 3.

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