Using a mathematical identity to speed up certain calculations.

[Maven]

The timings you requested, first "GaussLCM", then naive:

In [110]: %timeit sum_quotients5([2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43], 100000)
10 loops, best of 3: 93.9 ms per loop

In [111]: %timeit sum_quotients4([2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43], 100000)
10 loops, best of 3: 131 ms per loop

I had actually run the first one earlier, and had compared the results to your GaussLCM result in #36, which took 51 ms, hence my "surprisingly close" comments. I'm curious why Python seems so much slower with the Naive method than VB6. I tried a few minor variations and they took essentially the same amount of time as my original.

That said, I think this type of comparison is mostly an intellectual exercise. I wouldn't use Python in an important, speed-critical application; I'd use Cython or C. The main advantage of Python in this case in my mind is its simplicity, like the naive method being a single line; the fact that gcd, arbitrary-length integers, and more powerful iteration are built-in, no extra debugging or thought needed; and the overall "readability" of the code. I'd say efficiency isn't just a measurement of how fast the code runs; it also includes how long it took to correctly write and how much effort it cost the coder. For solving Project Euler's first problem, I'd say Python wins by a mile on that front: just type "sum(n for n in range(1000) if n%3 == 0 or n%5 == 0)" into an interpreter and you have your answer immediately.

I don't think one could tell much about comparing times like these. Processes are frequently swapped in and out by the operating system; as a result, one can't tell by looking at the system time how long exactly a process took to complete. Of course, this is outside of computer or operating system comparisons.

In addition, ask yourself what your trying to optimize for exactly. You're going to get different problems with resulting strategies depending on how you answer the question.