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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?
My 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.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 10" | ./5
Output:
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)
My code
… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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 rarely 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:
C# www.mathblog.dk/project-euler-problem-5/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/1-9/problem5.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p005.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/5 Smallest multiple.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem005.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p005.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p005.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler005.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/005-Smallest-multiple.pl (written by Gustaf Erikson)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
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