Problem:
Score : 400 points

Problem Statement

You are given positive integers N and M.

How many sequences a of length N consisting of positive integers satisfy a_1 \times a_2 \times ... \times a_N = M? Find the count modulo 10^9+7.

Here, two sequences a' and a'' are considered different when there exists some i such that a_i' \neq a_i''.

Constraints

All values in input are integers.

1 \leq N \leq 10^5

1 \leq M \leq 10^9

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of the sequences consisting of positive integers that satisfy the condition, modulo 10^9 + 7.

Sample Input 1

2 6

Sample Output 1

4

Four sequences satisfy the condition: \{a_1, a_2\} = \{1, 6\}, \{2, 3\}, \{3, 2\} and \{6, 1\}.

Sample Input 2

3 12

Sample Output 2

18

Sample Input 3

100000 1000000000

Sample Output 3

957870001

Solution in Kotlin (1.0.0) (judge verdict: Accepted)

import java.util.*

fun main(args: Array<String>) {
    val (N, M) = readLine()!!.split(" ").map {it.toLong()}
    val MOD = 1e9.toLong() + 7

    val fact = factorize(M)
    val comb = ModCombination(N + 100, MOD)

    println(
        fact.map {comb[N + it.second - 1, N - 1]}
            .fold(ModInt(1L, MOD)) { a, b -> a * b }
            .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 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) {
            return ModInt(0, mod)
        }

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

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))
}