Project Euler Problem 5: Smallest Multiple¶
The source code for this problem can be found here.
Problem Statement¶
\(2520\) 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 \(20\)?
Solution Discussion¶
\(2520\) is the smallest multiple of all numbers \(1\) through \(10\), which means that any multiple of \(2520\) is also divisible by \(1\) through \(10\). Any nonmultiple will not be divisible by ALL numbers \(1\) through \(10\), so can be ignored.
Divisibility by the numbers \(11\) through \(20\) should be tested from highest (\(20\)) to lowest (\(11\)) since higher divisors will rule more candidates out by indivisibility. More explicitly, \(19\) out of every \(20\) integers are not divisible by \(20\) whereas only \(18\) out of every \(19\) integers are not divisible by \(19\). This is akin to lazy boolean logic evaluation and avoids redundant computation.
Using these insights, a simple search strategy will find the answer very quickly. More specifically, search through increasing multiples of \(2520\) testing for divisibility by \(20,19,\dots,11\)  in that order. Identify the first, and thus smallest such number.
Solution Implementation¶
from typing import List
def is_multiple(n: int, divisors: List[int]) > bool:
""" Check whether :math:`n` is divisible by all of the given :math:`divisors`
:param n: the integer to check for divisibility
:param divisors: the divisors to test :math:`n` with
:return: whether :math:`n` is divisible by all :math:`divisors` or not
"""
for m in divisors:
if n % m != 0:
return False
return True
def solve():
""" Compute the answer to Project Euler's problem #5 """
divisors = [20, 19, 18, 17, 16, 15, 14, 13, 12, 11] # reverse order
n = 2520 # start our search at 2520
while not is_multiple(n, divisors):
n += 2520 # increment our search by 2520 at a time
return n
expected_answer = 232792560

solutions.problem5.
is_multiple
(n, divisors)¶ Check whether \(n\) is divisible by all of the given \(divisors\)
Parameters:  n (
int
) – the integer to check for divisibility  divisors (
List
[int
]) – the divisors to test \(n\) with
Return type: bool
Returns: whether \(n\) is divisible by all \(divisors\) or not
 n (

solutions.problem5.
solve
()¶ Compute the answer to Project Euler’s problem #5