2006 JBMO Problems/Problem 1
Problem
If is a composite number, then divides .
Solution
We shall prove a more stronger result that divides which will cover the case of the problem statement.
Let where .
Let us define set
First let's note that
Now, all multiples of from to
Since we have that Also, since we have that
So, we have that , in other words, divides
Now, all multiples of from to
Since we have that
Also, since so we have that
So, we have that , in other words, divides
Thus divides .