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// 中国剰余定理
// x = b_1 (mod_1), ..., x = b_k (mod_k), ... を満たす x を
// 0 <= x < lcm(mod_1, mod_2, ..., mod_k, ...) の範囲で求める
template <typename NumType, NumType LIMIT = NumType(1e10)>
struct CRT {
pair<NumType, NumType> NIL;
CRT() : NIL(-1, -1) {}
NumType extgcd(NumType a, NumType b, NumType &p, NumType &q) {
if(b == 0) {
p = 1, q = 0;
return a;
}
NumType d = extgcd(b, a%b, q, p);
q -= a / b * p;
return d;
}
pair<NumType, NumType> solve(NumType b1, NumType mod1, NumType b2, NumType mod2) {
NumType p, q;
NumType d = extgcd(mod1, mod2, p, q);
if((b2 - b1) % d != 0) return NIL;
NumType s = (b2 - b1) / d;
NumType t = (s * p % (mod2 / d));
// get lcm
NumType lc = mod1 / d * mod2;
NumType so = (b1 + mod1 * t) % lc;
(so += lc) %= lc;
return make_pair(so, lc);
}
// m, mod の vector をもらって、
// CRT の解を (x, lcm(mod_1, mod_2, ..., mod_k, ...)) の形で返す
pair<NumType, NumType> solve(vector<NumType> m, vector<NumType> mod) {
assert(m.size() == mod.size());
NumType so = 0, lc = 1;
for(size_t i=0; i<m.size(); i++) {
tie(so, lc) = solve(so, lc, m[i], mod[i]);
if(so > LIMIT or so < 0) {
return NIL;
}
}
return make_pair(so, lc);
}
};
#line 1 "math/math_021_crt.cpp"
// 中国剰余定理
// x = b_1 (mod_1), ..., x = b_k (mod_k), ... を満たす x を
// 0 <= x < lcm(mod_1, mod_2, ..., mod_k, ...) の範囲で求める
template <typename NumType, NumType LIMIT = NumType(1e10)>
struct CRT {
pair<NumType, NumType> NIL;
CRT() : NIL(-1, -1) {}
NumType extgcd(NumType a, NumType b, NumType &p, NumType &q) {
if(b == 0) {
p = 1, q = 0;
return a;
}
NumType d = extgcd(b, a%b, q, p);
q -= a / b * p;
return d;
}
pair<NumType, NumType> solve(NumType b1, NumType mod1, NumType b2, NumType mod2) {
NumType p, q;
NumType d = extgcd(mod1, mod2, p, q);
if((b2 - b1) % d != 0) return NIL;
NumType s = (b2 - b1) / d;
NumType t = (s * p % (mod2 / d));
// get lcm
NumType lc = mod1 / d * mod2;
NumType so = (b1 + mod1 * t) % lc;
(so += lc) %= lc;
return make_pair(so, lc);
}
// m, mod の vector をもらって、
// CRT の解を (x, lcm(mod_1, mod_2, ..., mod_k, ...)) の形で返す
pair<NumType, NumType> solve(vector<NumType> m, vector<NumType> mod) {
assert(m.size() == mod.size());
NumType so = 0, lc = 1;
for(size_t i=0; i<m.size(); i++) {
tie(so, lc) = solve(so, lc, m[i], mod[i]);
if(so > LIMIT or so < 0) {
return NIL;
}
}
return make_pair(so, lc);
}
};