PC3-12800 1600 MHz RAM is faster than PC3-10600 1333 MHz RAM, but that’s not relevant to your problem. The RAM bus (the pipeline connecting your RAM to your CPU) is 1333 MHz; faster RAM won’t make your computer any faster, because the CPU can’t talk to the RAM any faster than it’s designed to do.
RAM tends to be downward-compatible; faster RAM typically will scale down to a slower bus speed, as long as it’s the same type of RAM (PC2/PC3/PC4 etc.) and fits in the same socket. But the RAM scales down to the speed of the computer - the computer doesn’t speed up to the bus speed of the RAM. If you suspect you might be buying a newer PC3-12800 computer in the foreseeable future (like a 2012 mini, or a 2012 MacBook Pro), then it might make sense to get the faster RAM, so you have the option of moving it to the newer computer later.
I suggest you take the RAM you’ve already bought, install it in your mini, and see if it works. If it does, then enjoy it and move on. RAM is a commodity: If it works, different RAM won’t work “better”, it may just be more reliable. The standard practice in the RAM business is to offer a lifetime warranty, so you’re covered in any case.
If it doesn’t work, then contact the Amazon seller for replacement, and maybe get the faster RAM just in case.
PC3-10600 8 GB RAM Chip
Installation de RAM dans le Mac mini mi-2011
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