1967 IMO Problems/Problem 3
Let be natural numbers such that is a prime greater than Let Prove that the product is divisible by the product .
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 .
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: 
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