Problem 5: Smallest multiple

(see projecteuler.net/problem=5)

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?

Algorithm

Basically we have to find the Least Common Multiple of 1,2,..,20 (abbreviated as lcm, see en.wikipedia.org/wiki/Least_common_multiple).
In general, the lcm of two numbers a and b can be computed as: lcm(a,b)=frac{ab}{gcd(a,b)}

gcd stands for the Greatest Common Divisor (see en.wikipedia.org/wiki/Greatest_common_divisor).

Euclid's algorithm (en.wikipedia.org/wiki/Euclidean_algorithm) produces the gcd in a recursive way:
gcd(a,0) = 0
gcd(a,b) = gcd(b mod a, a)

An iterative version in C++ consists of just a few lines, see my gcd function.

Now that we know how to determine lcm(a,b) there is a pretty easy way to do the same for lcm(x_1,x_2,x_3,...,x_n):
lcm(x_1,x_2,x_3,...,x_{n-1},x_n) = lcm(lcm(x_1,x_2,x_3,...,x_{n-1}), x_n)

Example:
lcm(1,2,3,4)
=lcm(lcm(1,2,3),4)
=lcm(lcm(lcm(1,2),3),4)
=lcm(lcm(2,3),4)
=lcm(6,4)
=12

Note

Wikipedia lists some interesting alternatives with different runtime behavior.
Especially the binary algorithm can be much faster (at the cost of more code).

By the way: the G++ compiler offers an intrinsic called __gcd() which may be faster on some systems.
I highly suspect it is based on the binary algorithm.

My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
 
// greatest common divisor
unsigned long long gcd(unsigned long long a, unsigned long long b)
{
while (a != 0)
{
unsigned long long c = a;
a = b % a;
b = c;
}
return b;
}
 
// least common multiple
unsigned long long lcm(unsigned long long a, unsigned long long b)
{
// parenthesis avoid overflow
return a * (b / gcd(a, b));
}
 
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
 
// find least common multiple of all numbers
unsigned long long result = 1;
for (unsigned int i = 2; i <= x; i++)
result = lcm(result, i);
 
std::cout << result << std::endl;
}
return 0;
}

This solution contains 5 empty lines, 4 comments and 1 preprocessor command.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 10" | ./5

Output:

(please click 'Go !')

Note: the original problem's input 20 cannot be entered
because just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

Changelog

February 23, 2017 submitted solution
March 28, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler005

My code solves 4 out of 4 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as medium.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

Links

projecteuler.net/thread=5 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-problem-5/ (written by Kristian Edlund)
Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p005.hs (written by Nayuki)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p005.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p005.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem5.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem005.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/5 Smallest multiple.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler005.scala (written by Michael Bayne)

Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
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226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250
The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
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