<< 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?

# 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 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:*

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

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 |

76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 |

126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 |

151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 |

176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 |

201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 | 225 |

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 |

My username at Project Euler is

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

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