Thank you for your help!!
Here's the question:
Find all positive integers n divisible by all positive integers m where m is less than or equal to n^(1/2)
First note that the only which fill the condition are
. We'll see there are no others.
Suppose fills the conditions, and let .
We can suppose now that and then arrive at a contradiction.
, so there exists a positive integer such that
Therefore and we can write
That is, .
Write . Plugging in the previous equality yields:
And since we have , so
Thus where we denote
We have and and these numbers
are coprime so . Furthermore,
and is coprime to , and ,
so , and thus
contradicting our assumption that (see head of the post).