Problem in archievements after game add

good afternoon … When I add the games through the game adder and unlock the achievements, the system does not recognize it;
I noticed that I have to buy the game

While I don’t have and never have owned an Xbox 360, yes I’d imagine you will need to own the game in order to mod the achievements on your profile.
Think about it - what if Microsoft noticed you’re getting achievements for a game that there is absolutely no record of you ever playing on your account around the time you got them?