Well Nick mentioned earlier today in the office that he had thought of a quicker way to accomplish the first problem. I’m not sure if this is what he meant, but what I came up with is roughly this: you don’t need to run past all the multiples of each number (3 and 5 in our case) since you can run over the lcm instead. There will be a fixed amount added each time you pass a multiple of the lcm.

So for example with the numbers 3 and 5 chosen as the divisors we’re interested in we could organize the multiples into a list as follows:

- 3 + 5 + 6 + 9 + 10 + 12 +15 = 60
- 18 + 20 + 21 + 24 + 25 + 27 + 30 = 165
- 33 + 35 + 36 + 39 + 40 + 42 + 45 = 270
- . . .

You’ll notice that the difference between each sum here is 105 = 7 * 15 which is the number of divisible numbers below the lcm times the lcm. Well, you probably get the idea now . . . the only snag is the fudge factor: since 15*66 = 990 we need to pick up the last couple multiples above this guy before we get to 1000 (namely 993,996,999).

It seemed like a good idea to allow as many different dividing factors as you like at the same time, so I rewrote both mine and Nick’s solution to take an arbitrary list of numbers as an argument. For low values like 1000, the lcm idea doesn’t add much. But for higher maximum values and more factors, it’s a considerable speedup.

Here are some timing results for 3 and 5:

- Max = 1000, Lcm = .00083, All = .00081
- Max = 100000 Lcm = .12, All = .12
- Max = 10000000 Lcm = 3.1, All = 6.3

Now for 3,5 and 7:

- Max = 1000, Lcm = .00014, All = .001
- Max = 100000, Lcm = .012, All = .14
- Max = 10000000, Lcm = .51, All = 9.5

You can see that the Lcm method actually improves with more factors since we’re skipping ahead faster. Here’s the (kind of messy) code:

def solution2(divisors = [3,5,7], m_val = max_val):

nums_lcm = lcm(divisors)

base_set = set([nums_lcm])

for j in range(len(divisors)):

cur_div = divisors[j]

base_set |= set( cur_div * i for i in range(1, nums_lcm / cur_div) )

# sum of all numbers below least common multiple

base_sum = sum(base_set)

# of summands in the previous example

num_summands = len(base_set)

reps = max_val//nums_lcm

scale = num_summands * nums_lcm

big_skip = nums_lcm * reps

fudge_set = set( i + big_skip for i in base_set if i + big_skip < max_val )
fudge = sum(fudge_set)
t = time()
my_set = set( base_sum + (i*scale) for i in xrange(0,reps))
print_answer(sum(my_set) + fudge, time()-t)
[/sourcecode]
Note that I don't start the clock until after doing the setup calculations since technically this information would be given to us as part of specifying which factors to use.
Finally, here is some number magic: the answers for higher maximum values in the case of the factors 3 and 5 are as follows:

- 1000 => 233168
- 10000 => 23331668
- 100000 => 2333316668
- 1000000 => 233333166668

Holy shit, no?