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project euler problem 03

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So i solved the problem and was looking through other solutions and found this:

n = 600851475143
i = 2
while i*i < n:
    while n%i == 0:
        n = n / i
    i = i + 1
print ('answer is', n)

I don't really get why this solution works. Can anyone explain this to me?


BTW: This is my solution in c++:

#include <iostream>
#include <dinput.h>
#include <vector>

using namespace std;

long long largestpfactor(long long num) {
	double start = sqrt(num);

	for(int i = start; i > 1; i--) {
		if((num % i) == 0) {
			long long div = 2;

			while(((i % div) != 0) && div < i) {

			if(div == i) return i;

	return NULL;

int main() {

	cout << largestpfactor(600851475143) << endl;

	while(!GetAsyncKeyState(VK_END)) {


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Imagine n is formed by some primes, like


2*2*2*2*3*7*7*11 (or something like that?)


Now, what the algorithm does, is start from the smallest prime (2) to get them out by dividing as many times as possible.




Then it increments i by 1, so it divides by 3 so that goes out too.




now, it will divide by 4, which is not a prime. These non-prime numbers will however have no effect, because we already took all the 2's out and 4 is 2*2 (if its not divisible by 2, it certainly wont be by 4)


then we finally get to 7, and after dividing by 7 twice we are left with 11 which is the answer.

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