// >>> TEMPLATES #include using namespace std; using ll = long long; using ld = long double; using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; #define int ll #define rep(i, n) for (int i = 0; i < (int)(n); i++) #define rep1(i, n) for (int i = 1; i <= (int)(n); i++) #define repR(i, n) for (int i = (int)(n)-1; i >= 0; i--) #define rep1R(i, n) for (int i = (int)(n); i >= 1; i--) #define loop(i, a, B) for (int i = a; i B; i++) #define loopR(i, a, B) for (int i = a; i B; i--) #define all(x) begin(x), end(x) #define allR(x) rbegin(x), rend(x) #define rng(x, l, r) begin(x) + (l), begin(x) + (r) #define pb push_back #define eb emplace_back #define fst first #define snd second template constexpr auto mp(A &&a, B &&b) { return make_pair(forward(a), forward(b)); } template constexpr auto mt(T&&... x) { return make_tuple(forward(x)...); } template auto constexpr inf_ = numeric_limits::max()/2-1; auto constexpr INF32 = inf_; auto constexpr INF64 = inf_; auto constexpr INF = inf_; #ifdef LOCAL #include "debug.hpp" #else #define dump(...) (void)(0) #define say(x) (void)(0) #define debug if (0) #endif template struct pque : priority_queue, Comp> { vector &data() { return this->c; } void clear() { this->c.clear(); } }; template using pque_max = pque>; template using pque_min = pque>; template ::value, int> = 0> ostream& operator<<(ostream& os, T const& a) { bool f = true; for (auto const& x : a) os << (f ? "" : " ") << x, f = false; return os; } template ::value, int> = 0> ostream& operator<<(ostream& os, const T (&a)[N]) { bool f = true; for (auto const& x : a) os << (f ? "" : " ") << x, f = false; return os; } template ())), class = typename enable_if::value>::type> istream& operator>>(istream& is, T &a) { for (auto& x : a) is >> x; return is; } template ostream& operator<<(ostream& os, pair const& p) { return os << p.first << " " << p.second; } template istream& operator>>(istream& is, pair& p) { return is >> p.first >> p.second; } struct IOSetup { IOSetup() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); } } iosetup; template struct FixPoint : private F { constexpr FixPoint(F&& f) : F(forward(f)) {} template constexpr auto operator()(T&&... x) const { return F::operator()(*this, forward(x)...); } }; struct MakeFixPoint { template constexpr auto operator|(F&& f) const { return FixPoint(forward(f)); } }; #define MFP MakeFixPoint()| #define def(name, ...) auto name = MFP [&](auto &&name, __VA_ARGS__) template struct vec_impl { using type = vector::type>; template static type make_v(size_t n, U&&... x) { return type(n, vec_impl::make_v(forward(x)...)); } }; template struct vec_impl { using type = T; static type make_v(T const& x = {}) { return x; } }; template using vec = typename vec_impl::type; template auto make_v(Args&&... args) { return vec_impl::make_v(forward(args)...); } template void quit(T const& x) { cout << x << endl; exit(0); } template constexpr bool chmin(T& x, U const& y) { if (x > (T)y) { x = (T)y; return true; } return false; } template constexpr bool chmax(T& x, U const& y) { if (x < (T)y) { x = (T)y; return true; } return false; } template constexpr auto sumof(It b, It e) { return accumulate(b, e, typename iterator_traits::value_type{}); } template int sz(T const& x) { return x.size(); } template int lbd(C const& v, T const& x) { return lower_bound(begin(v), end(v), x)-begin(v); } template int ubd(C const& v, T const& x) { return upper_bound(begin(v), end(v), x)-begin(v); } constexpr int64_t mod(int64_t x, int64_t m) { assert(m > 0); return (x %= m) < 0 ? x+m : x; } constexpr int64_t div_floor(int64_t x, int64_t y) { assert(y != 0); return x/y - ((x^y) < 0 and x%y); } constexpr int64_t div_ceil(int64_t x, int64_t y) { assert(y != 0); return x/y + ((x^y) > 0 and x%y); } constexpr int dx[] = { 1, 0, -1, 0, 1, -1, -1, 1 }; constexpr int dy[] = { 0, 1, 0, -1, 1, 1, -1, -1 }; constexpr int popcnt(ll x) { return __builtin_popcountll(x); } mt19937_64 seed_{random_device{}()}; template Int rand(Int a, Int b) { return uniform_int_distribution(a, b)(seed_); } i64 irand(i64 a, i64 b) { return rand(a, b); } // [a, b] u64 urand(u64 a, u64 b) { return rand(a, b); } // template void shuffle(It l, It r) { shuffle(l, r, seed_); } template vector &operator--(vector &v) { for (T &x : v) --x; return v; } template vector &operator++(vector &v) { for (T &x : v) ++x; return v; } // <<< // >>> matrix (array) template decltype(T::one()) semi_ring_one(signed) { return T::one(); } template constexpr T semi_ring_one(long) { return 1; } template struct Matrix { int n = N, m = M; array, N> a; Matrix() : a() {} Matrix(initializer_list> init) { size_t i = 0; for (auto ls : init) { size_t j = 0; for (auto const& x : ls) { a[i][j] = x; j++; } assert((int)j == m); i++; } assert((int)i == n); } array const& operator[](int i) const { assert(0 <= i); assert(i < n); return a[i]; } array & operator[](int i) { assert(0 <= i); assert(i < n); return a[i]; } bool operator==(Matrix const& x) const { if (n != x.n || m != x.m) return false; rep (i, n) rep (j, m) if (a[i][j] != x[i][j]) return false; return true; } bool operator!=(Matrix const& x) const { return !(*this == x); } Matrix operator+() const { return *this; } Matrix operator+(Matrix const& x) const { return Matrix(*this) += x; } Matrix& operator+=(Matrix const& x) { assert(n == x.n); assert(m == x.m); rep (i, n) rep (j, m) a[i][j] += x[i][j]; return *this; } template Matrix operator*(Matrix const& x) const { assert(m == x.n); Matrix ret; rep (i, n) rep (j, m) rep (k, x.m) ret[i][k] += a[i][j] * x[j][k]; return ret; } Matrix& operator*=(Matrix const& x) { auto res = (*this)*x; swap(a, res.a); return *this; } Matrix operator*(T const& c) const { return Matrix(*this) *= c; } Matrix& operator*=(T const& c) { rep (i, n) rep (j, m) a[i][j] *= c; return *this; } friend Matrix operator*(T const& c, Matrix const& x) { Matrix ret = x; rep (i, x.n) rep (j, x.m) ret[i][j] = c*x[i][j]; return ret; } static Matrix identity() { assert(N == M); Matrix ret; rep (i, N) ret[i][i] = semi_ring_one(0); return ret; } Matrix pow(ll k) const { assert(n == m); assert(k >= 0); Matrix v = *this, r = Matrix::identity(); for (; k > 0; k >>= 1, v *= v) if (k&1) r *= v; return r; } #if 1 Matrix operator-() const { Matrix x = *this; rep (i, n) rep (j, m) x[i][j] = -x[i][j]; return x; } Matrix& operator-=(Matrix const& x) { assert(n == x.n); assert(m == x.m); rep (i, n) rep (j, m) a[i][j] -= x[i][j]; return *this; } Matrix operator-(Matrix const& x) const { return Matrix(*this) -= x; } Matrix& operator/=(T const& c) { rep (i, n) rep (j, m) a[i][j] /= c; return *this; } Matrix operator/(T const& c) const { return Matrix(*this) /= c; } #endif friend istream& operator>>(istream& is, Matrix& x) { rep (i, x.n) rep (j, x.m) is >> x[i][j]; return is; } #ifdef LOCAL friend string to_s(Matrix const& x) { string ret; rep (i, x.n) { ret += "\n("; rep (j, x.m) ret += " " + to_s(x[i][j]); ret += " )"; } return ret += "\n"; } #endif }; // <<< // >>> runtime modint template class runtime_modint { using u32 = uint32_t; using i32 = int32_t; using i64 = int64_t; using M = runtime_modint; u32 x; struct barrett { uint32_t mod; uint64_t inv; barrett() : mod(0), inv(0) { } barrett(uint32_t mod) : mod(mod), inv((uint64_t)(-1) / mod + 1) { } uint32_t operator()(uint32_t a, uint32_t b) const { return operator()(uint64_t(a) * b); } uint32_t operator()(uint64_t a) const { const __uint128_t c = a; const uint64_t q = (c * inv) >> 64; uint32_t x = c - q * mod; if (mod <= x) x += mod; return x; } }; inline static barrett bar; public: static void set_mod(u32 new_mod) { bar = barrett(new_mod); } static int mod() { return bar.mod; } runtime_modint(i64 x = 0) { assert(mod() > 0); if (x < 0) { x = -x; this->x = bar(x); if (this->x > 0) this->x = mod() - this->x; } else { this->x = bar(x); } } i64 val() const { return x; } constexpr explicit operator i64() const { return x; } bool operator==(M const& r) const { return x == r.x; } bool operator!=(M const& r) const { return x != r.x; } M operator+() const { return *this; } M operator-() const { return M()-*this; } M& operator+=(M const& r) { i64 t = i64(x) + r.x; if (t >= mod()) t -= mod(); x = t; return *this; } M& operator-=(M const& r) { i64 t = i64(x) + mod()-r.x; if (t >= mod()) t -= mod(); x = t; return *this; } M& operator*=(M const& r) { x = bar(x, r.x); return *this; } M& operator/=(M const& r) { return *this *= r.inv(); } M operator+(M r) const { return M(*this) += r; } M operator-(M r) const { return M(*this) -= r; } M operator*(M r) const { return M(*this) *= r; } M operator/(M r) const { return M(*this) /= r; } friend M operator+(i64 x, M y) { return M(x)+y; } friend M operator-(i64 x, M y) { return M(x)-y; } friend M operator*(i64 x, M y) { return M(x)*y; } friend M operator/(i64 x, M y) { return M(x)/y; } M pow(i64 n) const { // 0^0 = 1 if (n < 0) return inv().pow(-n); M v = *this, r = 1; for (; n > 0; n >>= 1, v *= v) if (n&1) r *= v; return r; } M inv() const { uint32_t a = x, b = mod(); int64_t u = 1, v = 0; while (b) { int64_t q = a / b; swap(a -= q * b, b); swap(u -= q * v, v); } assert(a == 1); return u; } static i64 gen() { // assume mod():prime if (mod() == 2) return 1; assert(mod() >= 3); for (int i = 2; i*i <= mod(); i++) assert(mod() % i != 0); vector ps; int n = mod()-1; for (int i = 2; i*i <= n; ++i) { if (n % i) continue; ps.push_back(i); do n /= i; while (n % i == 0); } if (n > 1) ps.push_back(n); n = mod()-1; auto check = [&](M g) { for (int p : ps) if (g.pow(n/p) == 1) return false; return true; }; for (int g = 2; g <= n; ++g) if (check(g)) return g; return -1; } // return minimum k >= (allow_zero ? 0 : 1) s.t. this->pow(k) == y // return -1 if not found int log(M y, bool allow_zero = false) { if (allow_zero and pow(0) == y) return 0; auto x = *this; M z = 1; int k = 0; while ((1u << k) < mod()) { z *= x, k++; if (z == y) return k; } u32 g = gcd(z.x, mod()); if (y.x % g != 0) return -1; auto old_bar = bar; bar = barrett(mod()/g); x.x %= mod(), y.x /= g, z.x /= g; unordered_map m; int s = 0; M w = 1; for ( ; s*s < mod(); s++) m[(y*w).x] = s, w *= x; while (k < mod()) { z *= w, k += s; if (m.count(z.x)) { swap(bar, old_bar); return k - m[z.x]; } } swap(bar, old_bar); return -1; } #ifdef LOCAL // friend string to_s(M r) { return to_s(r.val(), M::mod()); } friend string to_s(M r) { return to_s(r.val()); } #endif friend ostream& operator<<(ostream& os, M r) { return os << r.val(); } friend istream& operator>>(istream& is, M &r) { i64 x; is >> x; r = x; return is; } }; using mint = runtime_modint<-1>; // <<< int32_t main() { int n; cin >> n; vector c(10); rep1 (i, 9) cin >> c[i]; int g = 0; rep1 (i, 9) rep1 (j, 9) if (c[i] > 0 and c[j] > 0) { g = gcd(g, abs(i-j)*9); } dump(g); int kind = count_if(all(c), [](int num) { return num > 0; }); if (kind == 1) g = 1e9 + 7; mint::set_mod(g); using M = Matrix; auto A = M::identity(); rep1 (i, 9) { M B = { { 10, i }, { 0, 1 } }; A = B.pow(c[i]) * A; } int x = A[0][1].val(); dump(x); if (kind == 1) { cout << x << '\n'; } else { cout << gcd(g, x) << '\n'; } }