Download it from PM onto your new computer. Your code should work fine for the new computer and you will have the latest version. I did this a while back when I too bought a Mac Mini.
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In the Mac manual there should be a section on settings, program and file transfer from your old mac to your new mac. I've done it once. it requires a double ended usb cable or the use of an adapter. I used a firewire cable which is not supported anymore. My information is a few years old, things may have changed but I doubt.