算法

(暴力枚举)

注意到 $X_i$ 最大可以取到 $50$，而 $50$ 以内的素数只有 $15$ 个，一共有 $2^{15}$ 个可能的数，这个数字不是很大，所以直接按要求暴力枚举即可。

C++ 代码

#include <bits/stdc++.h>
#define rep(i, n) for (int i = 0; i < (n); ++i)

using std::cin;
using std::cout;
using std::min;
using std::gcd;
using std::vector;
using ll = long long;

const ll LINF = 1001002003004005006ll;

// linear sieve
vector<int> ps, pf;
void sieve(int mx) {
    pf.resize(mx + 1);
    rep(i, mx + 1) pf[i] = i; 
    for (int i = 2; i <= mx; ++i) {
        if (pf[i] == i) ps.push_back(i);
        for (int j = 0; j < ps.size() and ps[j] <= pf[i]; ++j) {
            int x = ps[j] * i;
            if (x > mx) break;
            pf[x] = ps[j];
        }
    }
}

int main() {
    sieve(50);

    int n;
    cin >> n;
    vector<int> a(n);
    rep(i, n) cin >> a[i];

    ll ans = LINF;
    int m = ps.size();
    int m2 = 1 << m;
    rep(i, m2) {
        ll x = 1;
        rep(j, m) if (i >> j & 1) x *= ps[j];
        bool ok = true;
        rep(j, n) if (gcd(x, a[j]) == 1) ok = false;
        if (ok) ans = min(ans, x);
    }

    cout << ans << '\n';

    return 0;
}