1967 IMO Problems/Problem 3
Contents
Problem
Let be natural numbers such that is a prime greater than Let Prove that the product is divisible by the product .
Solution
For any , .
We can therefore write the product in the problem as follows:
But, the product of any consecutive integers is divisible by . We can prove this as follows:
Therefore, is divisible by , and is divisible by . However, we are told that is prime and therefore it is not divisible by any of the numbers through . Therefore, is divisible by .
Finally, it is clear that is divisible by .
~mathboy100
Solution 2
We have that
and we have that
So we have that We have to show that:
is an integer
But is an integer and is an integer because but does not divide neither nor because is prime and it is greater than (given in the hypotesis) and .
The above solution was posted and copyrighted by Simo_the_Wolf. The original thread can be found here: [1]
See Also
1967 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |