Printing supplies can be very confusing because there are many ways of describing an item.

Part Number is the manufacturer's code for the item, and is generally fairly long, using 4 - 5 + characters.
Examples are 1384A004AA, C9497AN, 106R1047

Cartridge Number is an easier way to remember certain laser and ink products. Examples are HP 56, Lexmark 32, Epson 78, and Canon FX-8