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: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ABC152E
    {
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var AList = NNList(N);
            LIB_FactorizedNumber lcm = 1;
            foreach (var item in AList)
            {
                lcm = LIB_FactorizedNumber.LCM(lcm, item);
            }
            LIB_Mod sum = 0;
            LIB_Mod lcm_mod = lcm;
            for (var i = 0; i < N; i++)
            {
                sum += lcm_mod / AList[i];
            }
            Console.WriteLine(sum);
        }
        static class Console_
        {
            static Queue<string> param = new Queue<string>();
            public static string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static public void Main(string[] args) { if (args.Length == 0) { Console.SetOut(new Printer(Console.OpenStandardOutput())); } var t = new Thread(Solve, 134217728); t.Start(); t.Join(); Console.Out.Flush(); }
        static long NN => long.Parse(Console_.NextString());
        static double ND => double.Parse(Console_.NextString());
        static string NS => Console_.NextString();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => xorshift);
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_FactorizedNumber : IEquatable<LIB_FactorizedNumber>
    {
        LIB_Dict<long, long> dat;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FactorizedNumber(long x)
        {
            dat = new LIB_Dict<long, long>();
            foreach (var item in LIB_Math.Factors(x).GroupBy(e => e).ToDictionary(e => e.Key, e => e.LongCount())) dat[item.Key] = item.Value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FactorizedNumber(LIB_FactorizedNumber x)
        {
            dat = new LIB_Dict<long, long>();
            foreach (var item in x.dat) dat[item.Key] = item.Value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator LIB_FactorizedNumber(long x) => new LIB_FactorizedNumber(x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public explicit operator long(LIB_FactorizedNumber x) => x.dat.Aggregate(1L, (a, e) => a * LIB_Math.Pow(e.Key, e.Value));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator LIB_Mod(LIB_FactorizedNumber x) => x.dat.Aggregate((LIB_Mod)1, (a, e) => a * LIB_Mod.Pow(e.Key, e.Value));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Mul(LIB_FactorizedNumber x) { foreach (var item in x.dat) dat[item.Key] += item.Value; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Div(LIB_FactorizedNumber x) { foreach (var item in x.dat) dat[item.Key] -= item.Value; }
        static public bool operator ==(LIB_FactorizedNumber x, LIB_FactorizedNumber y) => x.dat.All(e => e.Value == y.dat[e.Key]) && y.dat.All(e => e.Value == x.dat[e.Key]);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator !=(LIB_FactorizedNumber x, LIB_FactorizedNumber y) => !(x == y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(LIB_FactorizedNumber x) => this == x;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object x) => x == null ? false : Equals((LIB_FactorizedNumber)x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode() => dat.Aggregate(0, (a, x) => a ^ x.Key.GetHashCode() ^ x.Value.GetHashCode());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_FactorizedNumber Pow(LIB_FactorizedNumber x, long y)
        {
            var ret = new LIB_FactorizedNumber(x);
            foreach (var item in ret.dat) ret.dat[item.Key] *= y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_FactorizedNumber GCD(LIB_FactorizedNumber x, LIB_FactorizedNumber y)
        {
            var ret = new LIB_FactorizedNumber(1);
            foreach (var item in x.dat) ret.dat[item.Key] = Min(item.Value, y.dat[item.Key]);
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_FactorizedNumber LCM(LIB_FactorizedNumber x, LIB_FactorizedNumber y)
        {
            var ret = new LIB_FactorizedNumber(x);
            var removeKeys = new HashSet<long>();
            foreach (var item in y.dat) ret.dat[item.Key] = Max(item.Value, x.dat[item.Key]);
            return ret;
        }
    }
    class LIB_Dict<K, V> : Dictionary<K, V>
    {
        Func<K, V> d;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Dict(Func<K, V> _d) { d = _d; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Dict() : this(_ => default(V)) { }
        new public V this[K i]
        {
            get
            {
                V v;
                return TryGetValue(i, out v) ? v : base[i] = d(i);
            }
            set
            {
                base[i] = value;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode()
        {
            var ret = 0;
            for (var e = GetEnumerator(); e.MoveNext();)
            {
                ret ^= e.Current.Key.GetHashCode();
                ret ^= e.Current.Value.GetHashCode();
            }
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object obj)
        {
            var o = (LIB_Dict<K, V>)obj;
            foreach (var kv in o)
            {
                if (!ContainsKey(kv.Key) || !this[kv.Key].Equals(kv.Value))
                    return false;
            }
            return o.Count == Count;
        }
    }
    struct LIB_Mod : IEquatable<LIB_Mod>, IEquatable<long>
    {
        static public long _mod = 1000000007; long v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Mod(long x)
        {
            if (x < _mod && x >= 0) v = x;
            else if ((v = x % _mod) < 0) v += _mod;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator LIB_Mod(long x) => new LIB_Mod(x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator long(LIB_Mod x) => x.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Add(LIB_Mod x) { if ((v += x.v) >= _mod) v -= _mod; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Sub(LIB_Mod x) { if ((v -= x.v) < 0) v += _mod; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Mul(LIB_Mod x) => v = (v * x.v) % _mod;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Div(LIB_Mod x) => v = (v * Inverse(x.v)) % _mod;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator +(LIB_Mod x, LIB_Mod y) { var t = x.v + y.v; return t >= _mod ? new LIB_Mod { v = t - _mod } : new LIB_Mod { v = t }; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator -(LIB_Mod x, LIB_Mod y) { var t = x.v - y.v; return t < 0 ? new LIB_Mod { v = t + _mod } : new LIB_Mod { v = t }; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator *(LIB_Mod x, LIB_Mod y) => x.v * y.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator /(LIB_Mod x, LIB_Mod y) => x.v * Inverse(y.v);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator ==(LIB_Mod x, LIB_Mod y) => x.v == y.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator !=(LIB_Mod x, LIB_Mod y) => x.v != y.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Inverse(long x)
        {
            long b = _mod, r = 1, u = 0, t = 0;
            while (b > 0)
            {
                var q = x / b;
                t = u;
                u = r - q * u;
                r = t;
                t = b;
                b = x - q * b;
                x = t;
            }
            return r < 0 ? r + _mod : r;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(LIB_Mod x) => v == x.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(long x) => v == x;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object x) => x == null ? false : Equals((LIB_Mod)x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode() => v.GetHashCode();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string ToString() => v.ToString();
        static List<LIB_Mod> _fact = new List<LIB_Mod>() { 1 };
        static long _factm = _mod;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void B(long n)
        {
            if (_factm != _mod) _fact = new List<LIB_Mod>() { 1 };
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i)
                    _fact.Add(_fact[i - 1] * i);
            _factm = _mod;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod Comb(long n, long k)
        {
            B(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod Perm(long n, long k)
        {
            B(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod Pow(LIB_Mod x, long y)
        {
            LIB_Mod a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a.Mul(x);
                x.Mul(x);
                y >>= 1;
            }
            return a;
        }
    }
    class LIB_Math
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Primes(long x)
        {
            if (x < 2) yield break;
            yield return 2;
            var halfx = x / 2;
            var table = new bool[halfx + 1];
            var max = (long)(Math.Sqrt(x) / 2);
            for (long i = 1; i <= max; ++i)
            {
                if (table[i]) continue;
                var add = 2 * i + 1;
                yield return add;
                for (long j = 2 * i * (i + 1); j <= halfx; j += add)
                    table[j] = true;
            }
            for (long i = max + 1; i <= halfx; ++i)
                if (!table[i] && 2 * i + 1 <= x)
                    yield return 2 * i + 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Factors(long x)
        {
            if (x < 2) yield break;
            while (x % 2 == 0)
            {
                x /= 2; yield return 2;
            }
            var max = (long)Math.Sqrt(x);
            for (long i = 3; i <= max; i += 2)
            {
                while (x % i == 0)
                {
                    x /= i; yield return i;
                }
            }
            if (x != 1) yield return x;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Divisor(long x)
        {
            if (x < 1) yield break;
            var max = (long)Math.Sqrt(x);
            for (long i = 1; i <= max; ++i)
            {
                if (x % i != 0) continue;
                yield return i;
                if (i != x / i) yield return x / i;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long GCD(long a, long b)
        {
            while (b > 0)
            {
                var tmp = b;
                b = a % b;
                a = tmp;
            }
            return a;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long LCM(long a, long b) => a / GCD(a, b) * b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Pow(long x, long y)
        {
            long a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a *= x;
                x *= x;
                y >>= 1;
            }
            return a;
        }
        static List<long> _fact = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void Build(long n)
        {
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Comb(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Perm(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<List<int>> MakePermutation(long n, bool zeroIndexed = true)
        {
            if (n <= 0) throw new Exception();
            var c = new int[n];
            var a = new int[n];
            if (!zeroIndexed) a[0] = 1;
            for (var i = 1; i < n; i++) a[i] = a[i - 1] + 1;
            yield return new List<int>(a);
            for (var i = 0; i < n;)
            {
                if (c[i] < i)
                {
                    if (i % 2 == 0)
                    {
                        var t = a[0]; a[0] = a[i]; a[i] = t;
                    }
                    else
                    {
                        var t = a[c[i]]; a[c[i]] = a[i]; a[i] = t;
                    }
                    yield return new List<int>(a);
                    ++c[i];
                    i = 0;
                }
                else
                {
                    c[i] = 0;
                    ++i;
                }
            }
        }
    }
}