Problem:
Score : 1400 points

Problem Statement

Count the number of strings S that satisfy the following constraints, modulo 10^9 + 7.

The length of S is exactly N.

S consists of digits (0...9).

You are given Q intervals.
For each i (1 \leq i \leq Q), the integer represented by S[l_i \ldots r_i] (the substring of S between the l_i-th (1-based) character and the r_i-th character, inclusive) must be a multiple of 9.

Here, the string S and its substrings may have leading zeroes.
For example, 002019 represents the integer 2019.

Constraints

1 \leq N \leq 10^9

1 \leq Q \leq 15

1 \leq l_i \leq r_i \leq N

Input

Input is given from Standard Input in the following format:

N
Q
l_1 r_1
:
l_Q r_Q

Output

Print the number of strings that satisfy the conditions, modulo 10^9 + 7.

Sample Input 1

4
2
1 2
2 4

Sample Output 1

136

For example, S = 9072 satisfies the conditions because both S[1 \ldots 2] = 90 and S[2 \ldots 4] = 072 represent multiples of 9.

Sample Input 2

6
3
2 5
3 5
1 3

Sample Output 2

2720

Sample Input 3

20
10
2 15
5 6
1 12
7 9
2 17
5 15
2 4
16 17
2 12
8 17

Sample Output 3

862268030

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Trace;
using SB = System.Text.StringBuilder;
using System.Numerics;
using static System.Math;
using N = System.Int64;
namespace Program {
    public class Solver {
        Random rnd = new Random(0);
        public void Solve() {
            var N = rl;
            var n = ri;
            var Q = Enumerate(n, x => new long[] { (rl - 1) * 100 + 32 + x, rl * 100 + 32 + x });
            var xs = Q.SelectMany(x => x).ToList();
            xs.Add(20);
            xs.Add(N * 100 + 71);
            xs = xs.Distinct().OrderBy(x => x).ToList();
            foreach (var a in Q) {
                a[0] = xs.BinarySearch(a[0]);
                a[1] = xs.BinarySearch(a[1]);
                //Debug.WriteLine(a.AsJoinedString());
            }
            var ten = new ModInt(10);
            var inv9 = ModInt.Inverse(9).num;
            var dist = new long[xs.Count, xs.Count];
            foreach (var p in Rep(xs.Count)) foreach (var q in Rep(xs.Count)) {
                    var i = (int)(xs[p] % 100);
                    var j = (int)(xs[q] % 100);
                    var d = Abs(xs[q] / 100 - xs[p] / 100);
                    dist[p, q] = (ModInt.Pow(ten, d).num + ModInt.Mod - 1) * inv9 % ModInt.Mod;
                }
            //Debug.WriteLine(inv9);
            var dp = Enumerate(1 << n, x => new long[9]);
            foreach (var i in Rep(1 << n)) foreach (var j in Rep(9)) dp[i][j] = -1;
            Func<int, int, int, int, int, long> calc = (s, t, p, q, v) => {
                var i = (int)(xs[p] % 100);
                var j = (int)(xs[q] % 100);
                var d = Abs(xs[q] / 100 - xs[p] / 100);
                if (j == 71) {
                    return ModInt.Pow(ten, d).num;
                } else if (j == 20 || (s >> j & 1) == 1) {
                    if (j == 20 && v != 0) return 0;
                    if (p > q) return 1;
                    return (dist[p, q] + 1);
                } else if ((t >> j & 1) == 1) {
                    return dist[p, q];
                } else return 1;
            };
            Func<int, int, long> dfs = null;
            dfs = (mask, v) => {
                if (mask == 0) return 1;
                if (v == 9) {
                    if (mask == 0) return 1;
                    else return 0;
                }
                if (dp[mask][v] != -1) return dp[mask][v];
                //Debug.WriteLine($"{mask} {v}");
                long ret = 0L;
                for (int s = mask; s >= 0; s = (s - 1) & mask) {
                    var rem = s ^ mask;
                    var add = dfs(rem, v + 1);
                    for (int u = 0; u < n; u++) {
                        if ((s >> u & 1) == 1) {
                            var l = (int)Q[u][0];
                            var r = (int)Q[u][1];
                            add = add * calc(s, rem, l, l - 1, v) % ModInt.Mod;
                            add = add * calc(s, rem, l, l + 1, v) % ModInt.Mod;
                            add = add * calc(s, rem, r, r - 1, v) % ModInt.Mod;
                            add = add * calc(s, rem, r, r + 1, v) % ModInt.Mod;
                        }
                    }
                    //Debug.WriteLine($"s:{s} rem:{rem} add:{add}");
                    ret += add;
                    if (s == 0) break;
                }
                //Debug.WriteLine($"{mask} {v}: {dp[mask][v]}");
                return dp[mask][v] = ret % ModInt.Mod;
            };
            Console.WriteLine(dfs((1 << n) - 1, 0));
        }
        const long INF = 1L << 60;
        int ri => sc.Integer();
        long rl => sc.Long();
        double rd => sc.Double();
        string rs => sc.Scan();
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static IEnumerable<int> Rep(int n) => Enumerable.Range(0, n);
        static IEnumerable<int> RRep(int n) => Enumerable.Range(0, n).Reverse();
        static T[] Enumerate<T>(int n, Func<int, T> f) {
            var a = new T[n];
            for (int i = 0; i < a.Length; ++i) a[i] = f(i);
            return a;
        }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex {
    static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
        return string.Join(st, ie);
    }
    static public void Main() {
        Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
        var solver = new Program.Solver();
        solver.Solve();
        Console.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO {
    using System.IO;
    using System.Text;
    using System.Globalization;

    public class Printer : StreamWriter {
        public override IFormatProvider FormatProvider => CultureInfo.InvariantCulture;
        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
    }

    public class StreamScanner {
        public StreamScanner(Stream stream) { str = stream; }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;

        private byte read() {
            if (isEof) return 0;
            if (ptr >= len) {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0) {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }

        public char Char() {
            byte b = 0;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }
        public string Scan() {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine() {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
        public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
        public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
}

#endregion

#region ModInt
/// <summary>
/// [0,<see cref="Mod"/>) までの値を取るような数
/// </summary>
public struct ModInt {
    /// <summary>
    /// 剰余を取る値．
    /// </summary>
    public const long Mod = (int)1e9 + 7;

    /// <summary>
    /// 実際の数値．
    /// </summary>
    public long num;
    /// <summary>
    /// 値が <paramref name="n"/> であるようなインスタンスを構築します．
    /// </summary>
    /// <param name="n">インスタンスが持つ値</param>
    /// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
    public ModInt(long n) { num = n; }
    /// <summary>
    /// このインスタンスの数値を文字列に変換します．
    /// </summary>
    /// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
    public override string ToString() { return num.ToString(); }
    public static ModInt operator +(ModInt l, ModInt r) { l.num += r.num; if (l.num >= Mod) l.num -= Mod; return l; }
    public static ModInt operator -(ModInt l, ModInt r) { l.num -= r.num; if (l.num < 0) l.num += Mod; return l; }
    public static ModInt operator *(ModInt l, ModInt r) { return new ModInt(l.num * r.num % Mod); }
    public static implicit operator ModInt(long n) { n %= Mod; if (n < 0) n += Mod; return new ModInt(n); }

    /// <summary>
    /// 与えられた 2 つの数値からべき剰余を計算します．
    /// </summary>
    /// <param name="v">べき乗の底</param>
    /// <param name="k">べき指数</param>
    /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
    public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }

    /// <summary>
    /// 与えられた 2 つの数値からべき剰余を計算します．
    /// </summary>
    /// <param name="v">べき乗の底</param>
    /// <param name="k">べき指数</param>
    /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
    public static ModInt Pow(long v, long k) {
        long ret = 1;
        for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
            if ((k & 1) == 1) ret = ret * v % Mod;
        return new ModInt(ret);
    }
    /// <summary>
    /// 与えられた数の逆元を計算します．
    /// </summary>
    /// <param name="v">逆元を取る対象となる数</param>
    /// <returns>逆元となるような値</returns>
    /// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
    public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
}
#endregion