# Understand the problem

Let be the set of positive integers. Determine all functions such that is divisible by for all positive integers .

##### Source of the problem

APMO 2019

##### Topic

Number Theory

##### Difficulty Level

Medium

##### Suggested Book

An Excursion in Mathematics

# Start with hints

Do you really need a hint? Try it first!

Considering prime numbers might be a good idea.

Show that, for any prime , . Hence or otherwise show that .

If for a prime , then show that . This is equivalent to . Note that the RHS is independent of .

As in the RHS is fixed and the LHS is arbitrarily large, the RHS has to be zero. Hence for every natural number .

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