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How was the Original Purchase made? The purchase that was for a FULL VERSION of Photoshop.
Was it done online, and downloaded? Did you make the purchase online, through an retailer, and then you were sent a disc in the mail? Or did you go to store, and purchase the disc?
The best thing would if you had made the purchase online....you would have a confirmation email about the purchase. And might be able to convince Adobe to give you a new product key.
But if you bought it from a store....all you would have is a paper trail in the form of a paper receipt, and if you lost it, and you lost the product keys, AND you lost the install discs......then you have lost all traces that the purchase of a FULL VERSION of photoshop was ever even made. Especially if you paid in cash.
Unless and However, you REGISTERED your FULL VERSION photoshop purchase, on the Adobe Website, then you might have a chance at retrieving the original product keys.
With the product keys, you can download a full version online, and then just activate photoshop with a product key.
Photoshop 5 is quite old so obtaining a newer version might be the best way to go. There may also be the possibility of finding a key for a prior version that someone is offering for sale. Often people will sell their old licenses after they upgrade. If you would like to pursue this option, reply back and I can offer you a few leads.