\[ \left\{\begin{array}{ccc} {xt_1} & {\equiv} & {a_{1}\pmod {p_{1}}} \\ {xt_2} & {\equiv} & {a_{2}\pmod {p_{2}}} \\ {} & {\vdots} & {} \\ {xt_n} & {\equiv} & {a_{k}\pmod {p_{n}}} \\ \end{array}\right. \]
\[ \left\{\begin{array}{ccc} {xt_1} & \geq & a_{1} \\ {xt_2} & \geq & a_{2} \\ {} & {\vdots} & \\ {xt_n} & \geq & a_{k} \\ \end{array}\right. \]
求出一个 \(x\) 满足上面两个方程组
首先用一次扩展欧几里得求出每个 \(x\) 的最小正整数解及其模数
再EXCRT合并,EXCRT的本质也是扩展欧几里得...
所以这是一道欧几里得算法练习题...
第二个不等式方程直接解就行
#include
typedef long long ll;
namespace IO {
const int MAXSIZE = 1 << 20;
char buf[MAXSIZE], *p1, *p2;
#define gc() (p1 == p2 && (p2 = (p1 = buf) + fread(buf, 1, MAXSIZE, stdin), p1 == p2) ? EOF : *p1++)
void read() {}
template
inline void read(T &x, T2 &... oth) {
x = 0; T f = 1; char ch = gc();
while (ch < '0' || ch > '9') { if (ch == '-') f = -1; ch = gc(); }
while (ch >= '0' && ch <= '9') { x = x * 10 + ch - 48; ch = gc(); }
x *= f;
read(oth...);
}
}
const int N = 1e5 + 7;
const ll INF = 0x3f3f3f3f3f3f3f3f;
std::multiset st;
ll a[N], r[N], p[N];
int n, m;
inline ll mul(ll a, ll b, ll mod) {
a %= mod;
if (a < 0) a += mod;
b %= mod;
if (b < 0) b += mod;
return (a * b - (ll)((long double)a / mod * b) * mod + mod) % mod;
}
ll exgcd(ll a, ll b, ll &x, ll &y) {
if (!b) {
x = 1, y = 0;
return a;
}
ll g = exgcd(b, a % b, x, y);
ll temp = x;
x = y;
y = temp - a / b * y;
return g;
}
void equ(ll a, ll b, ll c, ll &x, ll &y) {
ll g = exgcd(a, b, x, y);
if (c % g) {
x = y = -INF;
return;
}
a /= g, b /= g, c /= g;
x = (x % b + b) % b;
x = mul(x, c, b);
y = b;
}
bool merge(int i, int j) {
ll x, y;
equ(p[i], p[j], a[j] - a[i], x, y);
if (x == -INF) return 0;
a[j] = x * p[i] + a[i]; p[j] = y * p[i];
return 1;
}
int main() {
int T;
IO::read(T);
while (T--) {
st.clear();
IO::read(n, m);
for (int i = 1; i <= n; i++)
IO::read(a[i]);
for (int i = 1; i <= n; i++)
IO::read(p[i]);
for (int i = 1; i <= n; i++)
IO::read(r[i]);
for (int i = 1; i <= m; i++) {
ll x;
IO::read(x);
st.insert(x);
}
bool flag = 0;
ll least = 0;
for (int i = 1; i <= n; i++) {
std::multiset::iterator it = st.upper_bound(a[i]);
if (it != st.begin()) it--;
ll v = *it, x, y; st.erase(it);
st.insert(r[i]);
equ(v, p[i], a[i], x, y);
if (a[i] % v) least = std::max(least, a[i] / v + 1);
else least = std::max(least, a[i] / v);
//printf("%lld %lld %lld\n", v, x, y);
if (x == -INF) {
flag = 1;
break;
}
a[i] = x; p[i] = y;
if (i > 1) {
if (!merge(i - 1, i)) {
flag = 1;
break;
}
}
}
if (flag) {
puts("-1");
} else {
ll ans = a[n];
if (a[n] < least) {
least -= a[n];
if (least % p[n])
least = (least / p[n] + 1) * p[n];
ans = a[n] + least;
}
//printf("%lld %lld %lld\n", a[n], p[n], least);
printf("%lld\n", ans);
}
}
}