How can a 64KiB computer do the same task as a modern computer? I took an extremely interesting intro to CE course, learning about transistors/memory/logic gates up to LC-3 assembly. Something that was really interesting to me was learning about how a processor's frequency is its speed, determined by the clock signal (if anyone can point to a link of how a clock cycle is made/works or explain, highly appreciated) but my main question is about his claim that any turing-complete computer can accomplish the task of another - performing the same binary operations, just in different amounts of time - that the processor doesn't matter. This depends on memory, though, right? A modern video game wouldn't be able to run on a 64 KiB computer, so I'm wondering if when the textbook and my prof make that claim, it's simply referring to the frequency of the clock cycle (and assuming unlimited/uncapped memory). 
Thanks!
The key considerations are infinite memory and unlimited time.  If those two considerations were met, then all modern computers are equivalent.  Since no real computers have infinite memory, and no real humans can afford to wait forever for a calculation, that equivalence is more conceptual than real.