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| #include <bits/stdc++.h>
#define rep(i, l, r) for(int i = (l); i <= (r); i++)
#define per(i, r, l) for(int i = (r); i >= (l); i--)
#define mem(a, b) memset(a, b, sizeof a)
using namespace std;
using ll = long long;
ll mul(ll a, ll b, ll p) {
return (a * b - ll((long double)a / p * b + 0.5) * p + p) % p;
}
ll Pow(ll a, ll n, ll p, ll r = 1) {
for(; n; n >>= 1, a = mul(a, a, p))
if(n & 1) r = mul(r, a, p);
return r;
}
namespace Pollard_Rho {
int chk(ll n) {
for(ll a : {2, 3, 7, 61, 24251}) {
if(n == a) return 1;
if(Pow(a, n - 1, n) 1) return 0;
ll k = n - 1, t;
while(~k & 1) k >>= 1;
if((t = Pow(a, k, n)) == 1) continue;
while(t != 1 && t != n - 1) t = mul(t, t, n);
if(t != n - 1) return 0;
}
return 1;
}
ll f(ll x, ll c, ll p) { return (mul(x, x, p) + c) % p; }
ll PR(ll n) {
ll a = 0, b = 0, c = rand() % (n - 1) + 1, v = 1, g;
for(int k = 1; ; k <<= 1, a = b, v = 1) {
rep(i, 1, k) {
b = f(b, c, n), v = mul(v, abs(a - b), n);
if(!(i & 127) || i == k) {
g = __gcd(v, n);
if(g > 1) return g;
}
}
}
}
ll ans[100]; int cnt;
void solve(ll n) {
if(chk(n)) return void(ans[++cnt] = n);
ll d; do d = PR(n); while(d == n);
solve(d), solve(n / d);
}
void work() {
ll n;
while(cin >> n) {
cnt = 0;
if(chk(n)) { puts("Prime"); continue; }
solve(n), sort(ans + 1, ans + cnt + 1);
int t = 0;
rep(i, 1, cnt) {
if(ans[i] != ans[i - 1]) cout << ans[i];
t++; if(ans[i] != ans[i + 1]) {
if(t > 1) cout << '^' << t;
putchar(32), t = 0;
}
} putchar(10);
}
}
};
namespace Inv {
void exgcd(ll a, ll b, ll& d, ll& x, ll& y) {
if(b) exgcd(b, a % b, d, y, x), y -= a / b * x;
else d = a, x = 1, y = 0;
}
void inv(ll a, ll p) {
ll d, x, y; exgcd(a, p, d, x, y);
if(d > 1) puts("Non-existent!");
else cout << (x % p + p) % p;
}
void work() {
ll a, p;
while(cin >> a >> p) inv(a, p);
}
};
namespace Prime {
void work() {
ll n;
while(cin >> n) {
while(!Pollard_Rho::chk(n)) n++;
cout << n << endl;
}
}
};
namespace Cipolla {
ll n, p, II;
struct cmp {
ll r, i;
cmp operator *(const cmp& b) {
return {(r * b.r + i * b.i % p * II) % p, (r * b.i + i * b.r) % p };
}
} U = { 1, 0 };
int pow1(ll a, int n, ll r = 1) {
for(; n; n >>= 1, a = a * a % p)
if(n & 1) r = r * a % p;
return r;
}
cmp pow2(cmp a, int n, cmp r = U) {
for(; n; n >>= 1, a = a * a)
if(n & 1) r = r * a;
return r;
}
void work() {
while(cin >> n >> p) {
if(!n) { puts("0"); continue; }
if(p == 2) { cout << n << endl; continue; }
if(pow1(n, p / 2) != 1) { puts("Non-existent!"); continue; }
ll a;
do a = rand() % p, II = (a * a - n + p) % p; while(!a || pow1(II, p / 2) == 1);
int x1 = pow2({a, 1}, p / 2 + 1).r, x2 = p - x1;
if(x1 > x2) swap(x1, x2);
printf("%d %d\n", x1, x2);
}
}
};
int main() {
srand(time(0));
puts("Press 1 for integer factorization.");
puts("Press 2 to calculate the modular multiplicative inverse of a number.");
puts("Press 3 to find the first prime number greater than or equal to a number.");
puts("Press 4 to calculate the modular square root of a number.");
int op;
while(1) {
cin >> op;
if(op == 1) Pollard_Rho::work();
else if(op == 2) Inv::work();
else if(op == 3) Prime::work();
else if(op == 4) Cipolla::work();
else puts("Illegal input! Please re-enter your option.");
}
}
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