Keys are specific to the software that is installed, there is only one key that will fit your IDM and to get that key you must purchase a licence. The good key will then be e-mailed to you or will appear on the web page from which you paid for it.
Trying any other keys, thousands of them to find the one that will fit may take you to December 2099 if not more.
They made it that way to protect their copyright. Can you blame them ?