Primal number algorithm: works for small numbers (<100000 or so) and then fails

[RokSav]

I made a code to determine whether a number is prime or not. I doubt it's optimized, but that's not what's interesting. Thing is, this code works for numbers up to around 10^7, with 10^9 being described as a prime number which it obviously isn't. Thoughts on reason?

  #include "../std_lib_facilities.h" //include definitions

int main()
{
	int n = 1;
	int x = 2; //x starts at 2 because prime numbers are all div. by 1
	cout << "enter a number\n";
	cin >> n;
	if (n % 2 == 0 || n % 3 == 0) // number divisible by 2 or 3?
	{
		cout << "not prime number\n";
		keep_window_open();
		return 0;
	}

	else
	{
		if (n % 5 == 0 || n % 7 == 0) // number divisible by 5 or 7?
		{
			cout << "not prime number\n"; 
			keep_window_open();
			return 0;
		}
		else
		{
			while (x < n) //check if divisible by all numbers below
			{
				if (n % x == 0)
				{
					cout << "not prime number\n";
					keep_window_open();
					return 0;
				}
				else
				{
					x++;
				}
			}
			cout << "prime number\n";
			keep_window_open();
			return 0;
		}
	}
}

tl:dr for code: check if number divisible by 2 or 3, then by 5 or 7, then if number passes all those check all numbers beneath it. If it passes that, print that it's a prime number.

[Duthomhas]

It seems to be working for me.

C:\Users\Michael\Programming\cplusplus.com\random\p2>a
enter a number
1000000000
not prime number

C:\Users\Michael\Programming\cplusplus.com\random\p2>a
enter a number
1000000007
prime number

Have you considered any overflow problems?

BTW, you can significantly decrease the amount of time it takes by checking

while (x <= sqrt(n))
{

Do you have specific numbers for which it fails?

[RokSav]

Thank you for the pointing out the sqrt possibility, I forgot about that.
10^10 causes the prime number thing my bad didn't count the zeroes properly.
And I guess it's overflow-that explanation makes sense. Thank you.

[Duthomhas]

Yes, 10¹⁰ is overflow for int.

Since negative numbers cannot be prime, you might as well use unsigned long long int as your type. Then you'll be able to use numbers over 10¹⁸.

Hope this helps.