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 Kotlin (1.0.0) (judge verdict: Time Limit Exceeded)

import java.util.*

fun main(args: Array<String>) {
    val N = readLine()!!.toInt()
    val A = readLine()!!.split(" ").map{it.toLong()}
    val mp = TreeMap<Long, Long>()

    for (a in A) {
        val factors = factorize(a)

        for (p in factors) {
            mp[p.first] = Math.max(mp[p.first] ?: 0L, p.second)
        }
    }

    val MOD = 1e9.toLong() + 7
    var res = ModInt(0L, MOD)

    for (a in A) {
        val factorMap = factorize(a).toMap()
        var b = ModInt(1L, MOD)

        for (e in mp.entries) {
            b *= ModInt(e.key, MOD).pow((e.value - (factorMap[e.key] ?: 0L)))
        }

        res += b
    }

    println(res.value)
}

fun factorize(n: Long): List<Pair<Long, Long>> {
    val list = ArrayList<Pair<Long, Long>>()
    var m = n
    var k = 2L

    while (m > 1 && k * k <= m) {
        var s = 0L
        while (m % k == 0L) {
            m /= k
            s++
        }
        if (s > 0) {
            list.add(Pair(k, s))
        }
        k++
    }

    if (m > 1) {
        list.add(Pair(m, 1L))
    }

    return list
}

class ModInt (value: Long, val mod: Long) {
    init {
        if (mod <= 0) {
            throw IllegalArgumentException("ModInt cannot be created for non-positive modulus: $mod")
        }
    }

    val value = (value % mod + mod) % mod

    private fun assertSameMod(other: ModInt) {
        if (mod != other.mod) {
            throw IllegalArgumentException(
                "You cannot operate summation, subtraction and multiplication for ModInts with different moduli: $mod, ${other.mod}"
            )
        }
    }

    operator fun unaryMinus() = ModInt(mod - value, mod)

    operator fun plus(other: ModInt): ModInt {
        assertSameMod(other)

        return ModInt((value + other.value) % mod, mod)
    }

    operator fun plus(other: Long) = plus(ModInt(other, mod))

    operator fun minus(other: ModInt): ModInt {
        assertSameMod(other)

        return plus(-other)
    }

    operator fun minus(other: Long) = minus(ModInt(other, mod))

    operator fun times(other: ModInt): ModInt {
        assertSameMod(other)

        return ModInt((value * other.value) % mod, mod)
    }

    operator fun times(other: Long) = times(ModInt(other, mod))

    fun pow(k: Long): ModInt {
        var res = 1L
        var a = value
        var l = k

        while (l > 0) {
            if ((l and 1L) == 1L) {
                res = (res * a) % mod
            }

            a = a * a % mod
            l /= 2L
        }

        return ModInt(res, mod)
    }

    /**
     * Returns an inverse element of the value.
     * To ensure `m * m.inv() = 1`, the modulus should be prime, since it uses Fermat's little theorem.
     */
    fun inv() = pow(mod - 2)

    operator fun div(other: ModInt): ModInt {
        assertSameMod(other)
        if (other.value == 0L) {
            throw ArithmeticException()
        }

        return times(other.inv())
    }

    operator fun div(other: Long) = div(ModInt(other, mod))
}

class ModCombination(val n: Long, val mod: Long) {
    private val fact = Array(n.toInt() + 1) { ModInt(1, mod) }
    private val invfact = Array(n.toInt() + 1) { ModInt(1, mod) }

    init {
        for (i in 2..n.toInt()) {
            fact[i] = fact[i - 1] * i.toLong()
            invfact[i] = fact[i].inv()
        }
    }

    operator fun get(n: Long, m: Long): ModInt {
        if (m > n || n < 0 || m < 0) {
            return ModInt(0, mod)
        }

        return fact[n.toInt()] * invfact[m.toInt()] * invfact[(n - m).toInt()]
    }
}