Project Euler Prob Five | Python Fiddle

def prime_factors(n):
    """determines prime factorization of n (with n greater than 1), given as dictionary,
    where key is factor and value is number of times it occurs """
    factor_dict = {}
    while n > 1:
        for x in range(2, n+1):
            while n % x == 0:
                if x not in factor_dict:
                    factor_dict[x] = 1
                else:
                    factor_dict[x] += 1
                n /= x
    return factor_dict

def smallest(z):
    """determines the smallest positive number that is evenly divisible by all
    the numbers from 1 to z"""
    needed_factors = {}
    for i in range(2, z+1):
        factors = prime_factors(i)
        for key in factors:
            if key not in needed_factors:
                needed_factors[key] = factors[key]
            else:
                if factors[key] > needed_factors[key]:
                    needed_factors[key] = factors[key]
    answer = 1
    for key in needed_factors:
        times = needed_factors[key]
        for index in range(times):
            answer *= key
    return answer

print smallest(20)

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