1982 USAMO Problems/Problem 4
Contents
Problem
Prove that there exists a positive integer such that is composite for every integer .
Solution 1
Let be a prime number that divides and be a whole number less than such that If is a multiple of , then, for some integer , This simplifies to This implies that . Thus we conclude that there exists an integer such that is composite for all integral .
Solution 2
I claim that works
Consider the primes
Note that and that
Also,
Take to be an odd integer.
It is well known (and not hard to prove) that
Consider some cases:
When we have
When we have
When we have
When we have
When we have
When we have
When we have , since
And furthermore, so these numbers need be composite.
But this covers all cases; we are done
See Also
1982 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.