公式就是：

$$\gcd(\lvert x_2 - x_1 \rvert, \lvert y_2 - y_1 \rvert) + 1$$

AC code：

#include <bits/stdc++.h>
using namespace std;
#define ll long long

int main() {
    ll x1, y1, x2, y2;
    while (cin >> x1 >> y1 >> x2 >> y2) {
        ll dx = abs(x2 - x1);
        ll dy = abs(y2 - y1);
        ll g = __gcd(dx, dy);
        if (dx == 0 && dy == 0) {
            cout << 0 << endl;
        } else {
            cout << g - 1 << endl;
        }
    }
    return 0;
}

由数论知识可知，答案为（其中 $\gcd$ 表示最大公因数，此处 $\gcd(0,n)=n$）：

$$\gcd(|x_1-x_2|,|y_1-y_2|)+1$$

#include <bits/stdc++.h>
using namespace std;
#define y1 Y1
int main()
{
	int x1, y1, x2, y2;
	while (cin >> x1 >> y1 >> x2 >> y2)
	{
		int dx = abs(x1 - x2), dy = abs(y1 - y2);
        if (1ll * dx * dy == 0) cout << max(max(dx, dy) - 1, 0) << endl;
        else cout << __gcd(dx, dy) - 1 << endl;
	}
	return 0;
}