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 |