I use Memtest 86+; it is the best by far.
Memtest 86+ seems more well kept up-to-date than Memtest 86, but both are essentially the same.
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The 32-bit version vs. the 64-bit version doesn't matter much... They are the same program.
If you have more than 4 GB of RAM, to directly address it, you need 64-bit... That doesn't necessarily mean the 32-bit doesn't work around the problem using memory pages. Honestly, even if it does, it is sort of pointless because you are not likely to find a 32-bit machine with more than 4 GB of RAM... nor a 32 bit OS that can access more than 4 GB of memory.
If you have a 64-bit processor, use the 64-bit version. That will have you covered regardless of how much ram you have... Whether it be 256 MB or 256 GB RAM a 64-bit processor and the 64-bit software will handle it.
Yeah, the 32-bit Memtest 86+ WILL boot on a 64-bit system, but it would be inefficient and may not be reliable > 4 GB.
In contrast the 64-bit version won't boot on a 32-bit system.
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Solution
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Simply put: Use the 64-bit version of Memtest 86+ on any machine that will boot that memory tester.
If you have an old machine that won't boot Memtest 86+ x64, then run the legacy 32-bit version of Memtest 86+.
If the machine won't boot Memtest 86+ at all, try Memtest 86.