>I need to create algorithm (based on an integer number) that by applying some formula or equation to this number I can identify if the number belongs to group 1, 2, or 3 or any combination of them (e.g. 1 and 2, 1 and 3, 1 and 2 and 3, 2 and 3)
>For example, number 8 would indicate belonging to all three groups (since I can divide this number by the group number without a remainder.
>Any suggestions?

This will point you in the right direction:
http://www.jimloy.com/number/divis.htm
..of course 7 is tricky but this seems to work
http://www.maa.org/mathland/mathtrek_05_23_05.html