2003 Indonesia MO Problems/Problem 1
Problem
Prove that is divisible by for every integers .
Solutions
Solution 1
By Fermat's Little Theorem, so Also, so That means is divisible by and so is divisible by for every integer
Solution 2
The expression can be factored into In order to show that is divisible by we need to show that is divisible by and
Lemma 1: is divisible by
Obviously if since is a factor of the expression is divisible by If then making divisible by
Lemma 2: is divisible by
Obviously if since is a factor of the expression is divisible by If then making divisible by . Also, if then making divisible by .
Since we've shown that is divisible by and for all the value is divisible by
See Also
2003 Indonesia MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 2 |
All Indonesia MO Problems and Solutions |