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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?

# 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 solved **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 already 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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