Smallest Multiple

import time
time.process_time()

primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]       # Primes under 30, insert a prime generator (Euler Problem 7) to n if using n > 30

def primemultiple(n):                               # Based on primes only being divisible by themselves and 1, ...
    i = 0                                           #      all primes <= n multiplied together form a lower bound for smallest multiple
    minmultiple = 1
    while primes[i] <= n:
        minmultiple = minmultiple*primes[i]
        i += 1
    return minmultiple

'''
First implementation, realized a better implementation below

def smallestmultiple1ton(n):                        # Function to find the smallest value that divides all numbers 1 to n
    multiple = False
    min = primemultiple(n)
    small = min - min % n                           # Rounding the value off to the nearest lower n divisible value 
    # Rounding off to the nearest lower n allows making jumps of n size which in turn reduces the number of necessary test cases by a 
    # factor of n; it also ensures the values checked are already divisible by n, negating the necessity to check n divisibility each 
    # iteration
    while multiple == False:
        for i in range(n-1, 2, -1):                 # Checking if n-1 down to 2 for each candidate solution is divisible ...
            div = small / i                         #     not checking n every time reduced time to calculate from ~17.3 to ~13.4 seconds
            if div % 1 == 0:                        # Checking if an integer value after division
                multiple = True
            else:
                small += n
                multiple = False
                break
    return small
    
    
Second implementation realizes that the primemultiple function is not just a lower bound, but rather that the solution must, by the
properties of prime numbers, be a multiple of this number.

This realization reduced time to calculate from ~13.4 seconds to less than a tenth of a second'''

def smallestmultiple(n):
    multiple = False
    min = primemultiple(n)
    small = min
    while multiple == False:
        for i in range(n, 2, -1):
            div = small / i
            if div % 1 == 0:
                multiple = True
            else:
                small += min
                multiple = False
                break
    return small
    
print(smallestmultiple(20))                     # 1 to 20 is the Project Euler problem

print("\n", time.process_time(), "seconds")