<< problem 4 - Largest palindrome product | Sum square difference - problem 6 >> |

# 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.

# 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:

This is equivalent to`echo "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)*

# 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 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:

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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# 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.

<< problem 4 - Largest palindrome product | Sum square difference - problem 6 >> |