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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 ?

# 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:

This is equivalent to`echo "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)*

# 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 solved **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 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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