Problem: 2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Python Solution

This is a bit longer than it needs to be, but it should run rather quickly.

import time
# function to factor a given positive integer n
def factors(n):
    factors = []
    # remove any factors of 2 first
    while n % 2 == 0:
        n = n/2
    # now look for odd factors
    p = 3
    while n != 1:
        while n % p == 0:
            n = n/p
        p += 2
    return factors
# merge two lists of factors based on maximum multiplicities
def factor_append(factors,new):
    if len(factors) == 0: return new
    for i in range(len(new)):
        if i > 0 and new[i] == new[i-1]: continue
        new_count = new.count(new[i])
        old_count = factors.count(new[i])
        if new_count > old_count:
            for j in range(new_count - old_count): factors.append(new[i])
    return factors
# find the smallest positive evenly divisible number for a positive integer n
def smallest_evenly_divisible(num):
    F = []
    for i in range(1,num + 1):
        F = factor_append(F,factors(i))
    product = 1
    for i in F: product *= i
    return product
start = time.time()
product = smallest_evenly_divisible(20)
elapsed = (time.time() - start)
print "result %s returned in %s seconds." % (product,elapsed)

And, it does run rather quickly.

result 232792560 returned in 0.000357866287231 seconds.

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  1. [...] we come across the 10,001st prime. (This will be slow. We’ll make a faster version later.) In Project Euler Problem 5 we had a cheap (and inefficient) factorization routine. We’ll use that here. import time [...]