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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.InputMismatchException;
import java.util.Iterator;
import java.util.List;
import java.util.Map;
import java.util.NoSuchElementException;
import java.util.Objects;

public class Main {
    static final int MOD = 1_000_000_007;
    static final Mod mod = new Mod(MOD);

    static void solve(MyScanner in, MyWriter out) {
        int n = in.nextInt();
        long[] a = in.nextLongArray(n);
        List<List<PrimeFactor>> b = new ArrayList<>();
        for (long e : a) {
            if (e > 1) {
                b.add(PrimeFactor.factorize(e));
            } else {
                b.add(Arrays.asList(new PrimeFactor(2, 0)));
            }
        }
        Map<Long, Integer> m = new HashMap<>();
        for (List<PrimeFactor> l : b) {
            for (PrimeFactor f : l) {
                int exp = m.getOrDefault(f.primeNumber, 0);
                exp = Math.max(exp, f.exponent);
                m.put(f.primeNumber, exp);
            }
        }
        Mod.Value lcm = mod.value(1);
        for (Map.Entry<Long, Integer> e : m.entrySet()) {
            long num = e.getKey();
            long exp = e.getValue();
            lcm.multiply(mod.pow(num, exp));
        }
        Mod.Value v = mod.value(0);
        for (List<PrimeFactor> e : b) {
            Mod.Value v2 = mod.value(lcm.longValue());
            for (PrimeFactor f : e) {
                v2.div(mod.pow(f.primeNumber, f.exponent));
            }
            v.add(v2);
        }
        out.println(v);
    }
    public static final class PrimeFactor {
        public final long primeNumber;
        public final int exponent;

        PrimeFactor(long primeNumber, int exponent) {
            if (primeNumber <= 1) {
                throw new IllegalArgumentException();
            }
            if (exponent < 0) {
                throw new IllegalArgumentException();
            }
            this.primeNumber = primeNumber;
            this.exponent = exponent;
        }
        @Override
        public int hashCode() {
            return Objects.hash(exponent, primeNumber);
        }
        @Override
        public boolean equals(Object obj) {
            if (this == obj) {
                return true;
            }
            if (!(obj instanceof PrimeFactor)) {
                return false;
            }
            PrimeFactor other = (PrimeFactor)obj;
            return exponent == other.exponent
                   && primeNumber == other.primeNumber;
        }
        @Override
        public String toString() {
            return "PrimeFactor [primeNumber="
                   + primeNumber
                   + ", exponent="
                   + exponent
                   + "]";
        }
        // 素因数分解
        // O(√N)
        public static List<PrimeFactor> factorize(long n) {
            if (n <= 1) {
                throw new IllegalArgumentException();
            }
            List<PrimeFactor> result = new ArrayList<>();
            for (long i = 2; i * i <= n; i += (i == 2 ? 1 : 2)) {
                if (n % i != 0) {
                    continue;
                }
                int exp = 1;
                n /= i;
                for (; n % i == 0; n /= i) {
                    exp++;
                }
                result.add(new PrimeFactor(i, exp));
            }
            if (n != 1) {
                result.add(new PrimeFactor(n, 1));
            }
            return result;
        }
    }
    public static final class Mod {
        private final int divisor;
        public Mod(int divisor) {
            if (divisor <= 0) {
                throw new IllegalArgumentException();
            }
            this.divisor = divisor;
        }
        public long mod(long dividend) {
            return Math.floorMod(dividend, divisor);
        }
        public long add(long a, long b) {
            return mod(mod(a) + mod(b));
        }
        public long subtract(long a, long b) {
            return mod(mod(a) - mod(b));
        }
        public long multiply(long a, long b) {
            return mod(mod(a) * mod(b));
        }
        public long pow(long a, long b) {
            if (b < 0) {
                throw new IllegalArgumentException();
            }
            long r = 1;
            while (b > 0) {
                if ((b & 1) == 1) {
                    r = multiply(r, a);
                }
                a = multiply(a, a);
                b >>= 1;
            }
            return r;
        }
        public long inverse(long a) {
            // assert isCoprime(a, divisor);
            if (a == 0) {
                throw new IllegalArgumentException();
            }
            a = mod(a);
            long b = divisor;
            long u = 1;
            long v = 0;
            while (b != 0) {
                long q = a / b;
                a -= q * b;
                u -= q * v;

                long tmp = a;
                a = b;
                b = tmp;

                tmp = u;
                u = v;
                v = tmp;
            }
            return mod(u);
        }
        public long div(long a, long b) {
            return multiply(a, inverse(b));
        }
        public Value value(long initialValue) {
            return new Value(this, initialValue);
        }

        public static final class Value {
            private final Mod mod;
            private long value;
            private Value(Mod mod, long initialValue) {
                this.mod = mod;
                value = mod.mod(initialValue);
            }
            public long longValue() {
                return value;
            }
            public Value add(long a) {
                value = mod.add(value, a);
                return this;
            }
            public Value add(Value a) {
                return add(a.value);
            }
            public Value subtract(long a) {
                value = mod.subtract(value, a);
                return this;
            }
            public Value subtract(Value a) {
                return subtract(a.value);
            }
            public Value multiply(long a) {
                value = mod.multiply(value, a);
                return this;
            }
            public Value multiply(Value a) {
                return multiply(a.value);
            }
            public Value pow(long a) {
                value = mod.pow(value, a);
                return this;
            }
            public Value inverse() {
                value = mod.inverse(value);
                return this;
            }
            public Value div(long a) {
                value = mod.div(value, a);
                return this;
            }
            public Value div(Value a) {
                return div(a.value);
            }
            @Override
            public String toString() {
                return Long.toString(value);
            }
        }
    }

    public static void main(String[] args) {
        MyWriter w = new MyWriter(System.out);
        solve(new MyScanner(System.in), w);
        w.flush();
    }

    static final class MyScanner {
        static final int BUFFER_SIZE = 8192;
        private final InputStream in;
        private final byte[] buffer = new byte[BUFFER_SIZE];
        private int point;
        private int readLength;
        MyScanner(InputStream in) {
            this.in = in;
        }
        private byte readByte() {
            if (point < readLength) {
                byte result = buffer[point];
                point += 1;
                return result;
            }
            try {
                readLength = in.read(buffer);
            } catch (IOException e) {
                throw new AssertionError(null, e);
            }
            if (readLength == -1) {
                return -1;
            }
            point = 1;
            return buffer[0];
        }
        private static boolean isPrintableCharExceptSpace(byte c) {
            return 33 <= c && c <= 126;
        }
        public char nextChar() {
            byte c = readByte();
            while (!(c == -1 || isPrintableCharExceptSpace(c))) {
                c = readByte();
            }
            if (c == -1) {
                throw new NoSuchElementException();
            }
            return (char)c;
        }
        public String next() {
            return next(16);
        }
        public String next(int n) {
            byte c = readByte();
            while (!(c == -1 || isPrintableCharExceptSpace(c))) {
                c = readByte();
            }
            if (c == -1) {
                throw new NoSuchElementException();
            }
            StringBuilder b = new StringBuilder(n);
            do {
                b.append((char)c);
                c = readByte();
            } while (c != -1 && isPrintableCharExceptSpace(c));
            return b.toString();
        }
        public long nextLong() {
            byte c = readByte();
            while (!(c == -1 || isPrintableCharExceptSpace(c))) {
                c = readByte();
            }
            if (c == -1) {
                throw new NoSuchElementException();
            }
            boolean minus = false;
            if (c == '-') {
                minus = true;
                c = readByte();
            }
            long result = 0L;
            do {
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                result = result * 10L + (c - '0');
                c = readByte();
            } while (c != -1 && isPrintableCharExceptSpace(c));
            return minus ? -result : result;
        }
        public int nextInt() {
            long n = nextLong();
            if (n < Integer.MIN_VALUE || n > Integer.MAX_VALUE) {
                throw new InputMismatchException();
            }
            return (int)n;
        }
        public double nextDouble() {
            return Double.parseDouble(next());
        }
        public int[] nextIntArray(int n) {
            int[] result = new int[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextInt();
            }
            return result;
        }
        public long[] nextLongArray(int n) {
            long[] result = new long[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextLong();
            }
            return result;
        }
        public char[] nextCharArray(int n) {
            char[] result = new char[n];
            for (int i = 0; i < n; i++) {
                result[i] = nextChar();
            }
            return result;
        }
        public char[][] next2dCharArray(int n, int m) {
            char[][] result = new char[n][m];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < m; j++) {
                    result[i][j] = nextChar();
                }
            }
            return result;
        }
        public int[][] nextVerticalIntArrays(int arrayCount, int arrayLength) {
            int[][] result = new int[arrayCount][arrayLength];
            for (int i = 0; i < arrayLength; i++) {
                for (int j = 0; j < arrayCount; j++) {
                    result[j][i] = nextInt();
                }
            }
            return result;
        }
        public long[][] nextVerticalLongArrays(int arrayCount,
                                               int arrayLength) {
            long[][] result = new long[arrayCount][arrayLength];
            for (int i = 0; i < arrayLength; i++) {
                for (int j = 0; j < arrayCount; j++) {
                    result[j][i] = nextLong();
                }
            }
            return result;
        }
        public char[][] nextVerticalCharArrays(int arrayCount,
                                               int arrayLength) {
            char[][] result = new char[arrayCount][arrayLength];
            for (int i = 0; i < arrayLength; i++) {
                for (int j = 0; j < arrayCount; j++) {
                    result[j][i] = nextChar();
                }
            }
            return result;
        }
    }
    static final class MyWriter extends PrintWriter {
        MyWriter(OutputStream out) {
            super(out);
        }
        public void joinAndPrintln(int[] x) {
            joinAndPrintln(x, " ");
        }
        public void joinAndPrintln(int[] x, String delimiter) {
            StringBuilder b = new StringBuilder();
            if (x.length > 0) {
                b.append(x[0]);
                for (int i = 1; i < x.length; i++) {
                    b.append(delimiter).append(x[i]);
                }
            }
            println(b.toString());
        }
        public void joinAndPrintln(long[] x) {
            joinAndPrintln(x, " ");
        }
        public void joinAndPrintln(long[] x, String delimiter) {
            StringBuilder b = new StringBuilder();
            if (x.length > 0) {
                b.append(x[0]);
                for (int i = 1; i < x.length; i++) {
                    b.append(delimiter).append(x[i]);
                }
            }
            println(b.toString());
        }
        public void joinAndPrintln(Iterable<?> iterable) {
            joinAndPrintln(iterable, " ");
        }
        public void joinAndPrintln(Iterable<?> iterable, String delimiter) {
            StringBuilder b = new StringBuilder();
            for (Iterator<?> i = iterable.iterator(); i.hasNext();) {
                b.append(i.next());
                while (i.hasNext()) {
                    b.append(delimiter).append(i.next());
                }
            }
            println(b.toString());
        }
    }
}