Problem:
Score : 600 points

Problem Statement

Consider the following arithmetic progression with n terms:

x, x + d, x + 2d, \ldots, x + (n-1)d

What is the product of all terms in this sequence?
Compute the answer modulo 1\ 000\ 003.

You are given Q queries of this form.
In the i-th query, compute the answer in case x = x_i, d = d_i, n = n_i.

Constraints

1 \leq Q \leq 10^5

0 \leq x_i, d_i \leq 1\ 000\ 002

1 \leq n_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

Q
x_1 d_1 n_1
:
x_Q d_Q n_Q

Output

Print Q lines.

In the i-th line, print the answer for the i-th query.

Sample Input 1

2
7 2 4
12345 67890 2019

Sample Output 1

9009
916936

For the first query, the answer is 7 \times 9 \times 11 \times 13 = 9009.
Don't forget to compute the answer modulo 1\ 000\ 003.

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Linq.Expressions;
using System.IO;
using System.Text;
using System.Diagnostics;

using static util;
using P = pair<int, int>;

using Binary = System.Func<System.Linq.Expressions.ParameterExpression,
                           System.Linq.Expressions.ParameterExpression,
                           System.Linq.Expressions.BinaryExpression>;
using Unary = System.Func<System.Linq.Expressions.ParameterExpression,
                          System.Linq.Expressions.UnaryExpression>;

class Program {
    static StreamWriter sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
    static Scan sc = new Scan();
    const int M = 1000000007;
    const int M2 = 998244353;
    const long LM = 1L << 60;
    const double eps = 1e-11;
    static void Main(string[] args)
    {
        int M = 1000003;
        MyMath.Mod = M;
        int q = sc.Int;
        var fact = new long[M + 1];
        fact[0] = 1;
        for (int i = 1; i <= M; i++)
        {
            fact[i] = fact[i - 1] * i % M;
        }
        for (int _ = 0; _ < q; _++)
        {
            long x, d, n;
            sc.Multi(out x, out d, out n);
            if (d == 0) {
                Prt(MyMath.pow(x, n));
                continue;
            }
            if (x == 0) {
                Prt(0);
                continue;
            }
            x = x * MyMath.inv(d) % M;
            if (x + n - 1 >= M) {
                Prt(0);
                continue;
            }
            Prt(MyMath.pow(d, n) * fact[x + n - 1] % M * MyMath.inv(fact[x - 1]) % M);
        }
        sw.Flush();
    }

    static void DBG(string a) => Console.WriteLine(a);
    static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    static void DBG(params object[] a) => DBG(string.Join(" ", a));
    static void Prt(string a) => sw.WriteLine(a);
    static void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    static void Prt(params object[] a) => Prt(string.Join(" ", a));
}
class pair<T, U> : IComparable<pair<T, U>> {
    public T v1;
    public U v2;
    public pair() : this(default(T), default(U)) {}
    public pair(T v1, U v2) { this.v1 = v1; this.v2 = v2; }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) { a = v1; b = v2; }
    public static bool operator>(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) > 0;
    public static bool operator<(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) < 0;
    public static bool operator>=(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) >= 0;
    public static bool operator<=(pair<T, U> a, pair<T, U> b) => a.CompareTo(b) <= 0;
}
static class util {
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static T sq<T>(T a) => Operator<T>.Multiply(a, a);
    public static T Max<T>(params T[] a) => a.Max();
    public static T Min<T>(params T[] a) => a.Min();
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    static readonly int[] dd = { 0, 1, 0, -1 };
    static readonly string dstring = "RDLU";
    public static P[] adjacents(this P p) => adjacents(p.v1, p.v2);
    public static P[] adjacents(this P p, int h, int w) => adjacents(p.v1, p.v2, h, w);
    public static pair<P, char>[] adjacents_with_str(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new pair<P, char>(new P(i + dd[k], j + dd[k ^ 1]), dstring[k])).ToArray();
    public static pair<P, char>[] adjacents_with_str(int i, int j, int h, int w)
        => Enumerable.Range(0, dd.Length).Select(k => new pair<P, char>(new P(i + dd[k], j + dd[k ^ 1]), dstring[k])).Where(p => inside(p.v1.v1, p.v1.v2, h, w)).ToArray();
    public static P[] adjacents(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1])).ToArray();
    public static P[] adjacents(int i, int j, int h, int w)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1])).Where(p => inside(p.v1, p.v2, h, w)).ToArray();
    public static void Assert(bool cond) { if (!cond) throw new Exception(); }
    public static Dictionary<T, int> compress<T>(this IEnumerable<T> a)
        => a.Distinct().OrderBy(v => v).Select((v, i) => new { v, i }).ToDictionary(p => p.v, p => p.i);
    public static Dictionary<T, int> compress<T>(params IEnumerable<T>[] a) => compress(a.Aggregate(Enumerable.Union));
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}
static class Operator<T> {
    static readonly ParameterExpression x = Expression.Parameter(typeof(T), "x");
    static readonly ParameterExpression y = Expression.Parameter(typeof(T), "y");
    public static readonly Func<T, T, T> Add = Lambda(Expression.Add);
    public static readonly Func<T, T, T> Subtract = Lambda(Expression.Subtract);
    public static readonly Func<T, T, T> Multiply = Lambda(Expression.Multiply);
    public static readonly Func<T, T, T> Divide = Lambda(Expression.Divide);
    public static readonly Func<T, T> Plus = Lambda(Expression.UnaryPlus);
    public static readonly Func<T, T> Negate = Lambda(Expression.Negate);
    public static Func<T, T, T> Lambda(Binary op) => Expression.Lambda<Func<T, T, T>>(op(x, y), x, y).Compile();
    public static Func<T, T> Lambda(Unary op) => Expression.Lambda<Func<T, T>>(op(x), x).Compile();
}

class Scan {
    StreamReader sr;
    public Scan() { sr = new StreamReader(Console.OpenStandardInput()); }
    public Scan(string path) { sr = new StreamReader(path); }
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => sr.ReadLine().Trim();
    public pair<T, U> Pair<T, U>() {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P => Pair<int, int>();
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); }
    public void Multi<T, U, V>(out T a, out U b, out V c)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]); }
}

static class MyMath
{
    public static long Mod = 1000000007;
    public static bool isprime(long a) {
        if (a < 2) return false;
        for (long i = 2; i * i <= a; i++) if (a % i == 0) return false;
        return true;
    }
    public static bool[] sieve(int n) {
        var p = new bool[n + 1];
        for (int i = 2; i <= n; i++) p[i] = true;
        for (int i = 2; i * i <= n; i++) if (p[i]) for (int j = i * i; j <= n; j += i) p[j] = false;
        return p;
    }
    public static bool[] segmentSieve(long l, long r) {
        int sqn = (int)Math.Sqrt(r + 9);
        var ps = getprimes(sqn);
        var sieve = new bool[r - l + 1];
        for (long i = l; i <= r; i++) sieve[i - l] = true;
        foreach (long p in ps)
        {
            if (p * p > r) break;
            for (long i = p >= l ? p * p : (l + p - 1) / p * p; i <= r; i += p) sieve[i - l] = false;
        }
        return sieve;
    }
    public static bool[] segmentSieve(long l, long r, List<int> ps) {
        var sieve = new bool[r - l + 1];
        for (long i = l; i <= r; i++) sieve[i - l] = true;
        foreach (long p in ps)
        {
            if (p * p > r) break;
            for (long i = p >= l ? p * p : (l + p - 1) / p * p; i <= r; i += p) sieve[i - l] = false;
        }
        return sieve;
    }
    public static List<int> getprimes(int n) {
        var prs = new List<int>();
        var p = sieve(n);
        for (int i = 2; i <= n; i++) if (p[i]) prs.Add(i);
        return prs;
    }
    public static long pow(long a, long b, long mod) {
        a %= mod;
        if (a == 0) return 0;
        if (b == 0) return 1;
        var t = pow(a, b / 2, mod);
        if ((b & 1) == 0) return t * t % mod;
        return t * t % mod * a % mod;
    }
    public static long pow(long a, long b) {
        a %= Mod;
        if (a == 0) return 0;
        if (b == 0) return 1;
        var t = pow(a, b / 2);
        if ((b & 1) == 0) return t * t % Mod;
        return t * t % Mod * a % Mod;
    }
    public static long inv(long a) => pow(a, Mod - 2);
    public static long gcd(long a, long b) {
        while (b > 0) { var t = a % b; a = b; b = t; } return a;
    }
    // a x + b y = gcd(a, b)
    public static long extgcd(long a, long b, out long x, out long y) {
        long g = a; x = 1; y = 0;
        if (b > 0) { g = extgcd(b, a % b, out y, out x); y -= a / b * x; }
        return g;
    }

    // 中国剰余定理
    // リターン値を (r, m) とすると解は x ≡ r (mod. m)
    // 解なしの場合は (0, -1) をリターン
    public static pair<long, long> chineserem(IList<long> b, IList<long> m) {
        long r = 0, M = 1;
        for (int i = 0; i < b.Count; ++i) {
            long p, q;
            long d = extgcd(M, m[i], out p, out q); // p is inv of M/d (mod. m[i]/d)
            if ((b[i] - r) % d != 0) return new pair<long, long>(0, -1);
            long tmp = (b[i] - r) / d * p % (m[i]/d);
            r += M * tmp;
            M *= m[i]/d;
        }
        return new pair<long, long>((r % M + M) % M, M);
    }

    public static long lcm(long a, long b) => a / gcd(a, b) * b;

    static long[] facts, invs;
    public static void setfacts(int n) {
        facts = new long[n + 1];
        facts[0] = 1;
        for (int i = 1; i <= n; i++) facts[i] = facts[i - 1] * i % Mod;
        invs = new long[n + 1];
        invs[n] = inv(facts[n]);
        for (int i = n; i > 0 ; i--) invs[i - 1] = invs[i] * i % Mod;
    }
    public static long comb(long n, long r) {
        if (n < 0 || r < 0 || r > n) return 0;
        if (facts != null && facts.Length > n) return facts[n] * invs[r] % Mod * invs[n - r] % Mod;
        if (n - r < r) r = n - r;
        long numer = 1, denom = 1;
        for (long i = 0; i < r; i++) {
            numer = numer * ((n - i) % Mod) % Mod;
            denom = denom * ((i + 1) % Mod) % Mod;
        }
        return numer * inv(denom) % Mod;
    }
    public static long[][] getcombs(int n) {
        var ret = new long[n + 1][];
        for (int i = 0; i <= n; i++) {
            ret[i] = new long[i + 1];
            ret[i][0] = ret[i][i] = 1;
            for (int j = 1; j < i; j++) ret[i][j] = (ret[i - 1][j - 1] + ret[i - 1][j]) % Mod;
        }
        return ret;
    }
    // nC0, nC2, ..., nCn
    public static long[] getcomb(int n) {
        var ret = new long[n + 1];
        ret[0] = 1;
        for (int i = 0; i < n; i++) ret[i + 1] = ret[i] * (n - i) % Mod * inv(i + 1) % Mod;
        return ret;
    }
    public static bool nextPermutation<T>(IList<T> p) where T : struct, IComparable<T> {
        for (int i = p.Count - 2; i >= 0; --i) {
            if (p[i].CompareTo(p[i + 1]) < 0) {
                for (int j = p.Count - 1; ; --j) {
                    if (p[j].CompareTo(p[i]) > 0) {
                        p.swap(i, j);
                        for(++i, j = p.Count - 1; i < j; ++i, --j)
                            p.swap(i, j);

                        return true;
                    }
                }
            }
        }
        return false;
    }
    public static bool nextPermutation<T>(IList<T> p, Comparison<T> compare) where T : struct {
        for (int i = p.Count - 2; i >= 0; --i) {
            if (compare(p[i], p[i + 1]) < 0) {
                for (int j = p.Count - 1; ; --j) {
                    if (compare(p[j], p[i]) > 0) {
                        p.swap(i, j);
                        for (++i, j = p.Count - 1; i < j; ++i, --j)
                            p.swap(i, j);

                        return true;
                    }
                }
            }
        }
        return false;
    }
}