Get IDs of children of category X

Hi!
I thought it was an easy task but am not managing to solve it. All I want is to make a loop to display 5 posts for every child-category of a specific category. To do this, I guess I have to obtain the IDs of those categories but I can’t find out how to do this.

Here’s what I ant to do:
1. Obtain all IDs for children of category X;
2. For each child-category, create a Loop with get_posts;
3. Display information about the posts for each child category;

The thing is that these child-categories will be continuously updated, so I had to search for them automatically…