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Problem 3: Largest prime factor
(see projecteuler.net/problem=3)
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
My Algorithm
Each composite number x can be represented as the product of at least two factors: x=factor*other
If we assume that factor is a prime number and factor<=other, then there are two options:
1. other can be a prime number, too
2. other is composite
In case 1, other is the largest prime - and we are done.
In case 2, we can continue the same process by setting set x=other.
After some iterations we will hit case 1.
Therefore I start a loop beginning with factor=2 (the smallest prime)
and as long as our number x can be divided by factor with remainder 0:
- divide x by factor, but abort if x==factor because then we have found our largest prime factor.
Note
You may have noticed that factor isn't always a prime number in my program:
yes, I simply scan through all numbers, even composite ones.
But if they are composite, then I already checked all smaller primes.
That means, I checked all prime factors of that composite number, too.
Therefore x can't be divided by a composite factor with remainder 0 because
all required prime factors were already eliminated from x.
In short: those divisions by composite numbers always fail but my program is still fast enough and
writing a proper prime sieve doesn't give a significant speed boost for this problem.
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 987654321" | ./3
Output:
Note: the original problem's input 600851475143
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.
#include <iostream>
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned long long x;
std::cin >> x;
// x can be represented by x=factor*otherFactor
// where factor <= otherFactor
// therefore factor <= sqrt(x)
for (unsigned long long factor = 2; factor * factor <= x; factor++)
// remove factor, actually it's a prime
// (can occur multiple times, e.g. 20=2*2*5)
while (x % factor == 0 && x != factor) // note: keep last prime
x /= factor;
std::cout << x << std::endl;
}
return 0;
}
This solution contains 3 empty lines, 5 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 25, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler003
My code solves 6 out of 6 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (out of 100%).
Hackerrank describes this problem as easy.
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=3 - 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-3/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/1-9/problem3.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p003.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/3 Largest prime factor.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem003.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p003.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p003.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler003.scala (written by Michael Bayne)
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 |
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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 !!!
<< problem 2 - Even Fibonacci numbers | Largest palindrome product - problem 4 >> |