Problem:
Score : 500 points

Problem Statement

Given are N positive integers A_1,...,A_N.

Consider positive integers B_1, ..., B_N that satisfy the following condition.

Condition: For any i, j such that 1 \leq i < j \leq N, A_i B_i = A_j B_j holds.

Find the minimum possible value of B_1 + ... + B_N for such B_1,...,B_N.

Since the answer can be enormous, print the sum modulo (10^9 +7).

Constraints

1 \leq N \leq 10^4

1 \leq A_i \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the minimum possible value of B_1 + ... + B_N for B_1,...,B_N that satisfy the condition, modulo (10^9 +7).

Sample Input 1

3
2 3 4

Sample Output 1

13

Let B_1=6, B_2=4, and B_3=3, and the condition will be satisfied.

Sample Input 2

5
12 12 12 12 12

Sample Output 2

5

We can let all B_i be 1.

Sample Input 3

3
1000000 999999 999998

Sample Output 3

996989508

Print the sum modulo (10^9+7).

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

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

using E = System.Linq.Enumerable;
using System.Numerics;

internal partial class Solver {

    public void Run() {
        var n = ni();
        var a = nl(n);
        var hist = new Dictionary<long, int>();
        foreach (var x in a) {
            var ts = Factorization(x);
            foreach (var kv in ts) {
                if (!hist.ContainsKey(kv.Item1)) hist[kv.Item1] = 0;
                hist[kv.Item1] = Math.Max(hist[kv.Item1], kv.Item2);
            }
        }
        int mod = 1000000007;
        ModInt.SetModValue(mod);
        ModInt lcm = 1;
        foreach (var kv in hist) {
            lcm = lcm * ModInt.Pow(kv.Key, kv.Value);
        }
        ModInt ans = 0;
        foreach (var x in a) {
            ans += lcm / x;
        }
        cout.WriteLine(ans);
    }


    public struct ModInt {
        private static int mod = 0;
        private long value;

        public ModInt(long x) {
            value = x;
            Normalize();
        }

        private static long RegularMod(long x, int mod) {
            if (x >= mod) {
                if (x < 2 * mod) {
                    return x - mod;
                }

                return x % mod;
            }
            if (x >= 0) {
                return x;
            }

            x = mod - RegularMod(-x, mod);
            if (x == mod) {
                return 0;
            }

            return x;
        }

        public static int SetModValue(int m) {
            return mod = m;
        }

        private void Normalize() {
            value = RegularMod(value, mod);
        }

        public override string ToString() {
            return value.ToString(CultureInfo.InvariantCulture);
        }

        public int ToInt() {
            return (int)value;
        }

        public static bool operator ==(ModInt c1, ModInt c2) {
            return c1.value == c2.value;
        }

        public static bool operator !=(ModInt c1, ModInt c2) {
            return !(c1 == c2);
        }

        public static ModInt operator +(ModInt x, ModInt y) {
            return new ModInt(x.value + y.value);
        }
        public static ModInt operator -(ModInt x, ModInt y) {
            return new ModInt(x.value - y.value);
        }
        public static ModInt operator *(ModInt x, ModInt y) {
            return new ModInt(x.value * y.value);
        }
        public static ModInt operator /(ModInt x, ModInt y) {
            return new ModInt(x.value * Inverse(y.value, mod));
        }
        public static ModInt operator ++(ModInt x) {
            return x + 1;
        }
        public static ModInt operator --(ModInt x) {
            return x - 1;
        }
        public static ModInt Pow(ModInt x, long n) {
            ModInt r = 1;
            while (n > 0) {
                if ((n & 1) != 0) r *= x;
                x *= x;
                n >>= 1;
            }
            return r;
        }
        private static long ExtendedGcd(long a, long b, ref long x, ref long y) {
            if (b == 0) {
                x = 1; y = 0;
                return a;
            } else {
                long d = ExtendedGcd(b, a % b, ref y, ref x);
                y -= a / b * x;
                return d;
            }
        }

        private static long Inverse(long a, long mod) {
            long x = 0, y = 0;
            if (ExtendedGcd(a, mod, ref x, ref y) == 1) {
                return (x + mod) % mod;
            } else {
                throw new Exception("Invalid inverse " + a + " " + mod);
            }
        }

        public static implicit operator ModInt(long x) {
            return new ModInt(x);
        }

        public override bool Equals(object obj) {
            if (obj == null) {
                return false;
            }
            return value.Equals(((ModInt)obj).value);
        }

        public override int GetHashCode() {
            return value.GetHashCode();
        }
    }
    public static List<Tuple<long, int>> Factorization(long x) {
        var ret = new List<Tuple<long, int>>();
        for (int i = 2; i * i <= x; i++) {
            if (x % i == 0) {
                int cnt = 0;
                while (x % i == 0) {
                    x /= i;
                    cnt++;
                }
                ret.Add(Tuple.Create((long)i, cnt));
            }
        }
        if (x > 1) ret.Add(Tuple.Create(x, 1));

        return ret;
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return (int)NextLong(); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new InvalidOperationException();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}