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 ?

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.

We can abort as soon as all factor<=sqrt{x} are processed because then only a prime is left.

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.

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.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 987654321" | ./3

Output:

(please click 'Go !')

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)

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 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 never 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:

Python: www.mathblog.dk/project-euler-problem-3/ (written by Kristian Edlund)
Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p003.hs (written by Nayuki)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p003.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p003.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem3.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem003.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/3 Largest prime factor.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler003.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:

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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !