<p>You are given a 2D integer array, <code>queries</code>. For each <code>queries[i]</code>, where <code>queries[i] = [n<sub>i</sub>, k<sub>i</sub>]</code>, find the number of different ways you can place positive integers into an array of size <code>n<sub>i</sub></code> such that the product of the integers is <code>k<sub>i</sub></code>. As the number of ways may be too large, the answer to the <code>i<sup>th</sup></code> query is the number of ways <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>Return <em>an integer array </em><code>answer</code><em> where </em><code>answer.length == queries.length</code><em>, and </em><code>answer[i]</code><em> is the answer to the </em><code>i<sup>th</sup></code><em> query.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> queries = [[2,6],[5,1],[73,660]]
<strong>Output:</strong> [4,1,50734910]
<strong>Explanation:</strong>&nbsp;Each query is independent.
[2,6]: There are 4 ways to fill an array of size 2 that multiply to 6: [1,6], [2,3], [3,2], [6,1].
[5,1]: There is 1 way to fill an array of size 5 that multiply to 1: [1,1,1,1,1].
[73,660]: There are 1050734917 ways to fill an array of size 73 that multiply to 660. 1050734917 modulo 10<sup>9</sup> + 7 = 50734910.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> queries = [[1,1],[2,2],[3,3],[4,4],[5,5]]
<strong>Output:</strong> [1,2,3,10,5]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= queries.length &lt;= 10<sup>4</sup> </code></li>
	<li><code>1 &lt;= n<sub>i</sub>, k<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
</ul>

MOD_VALUE = int(1e9 + 7)

class Solution:
    def __init__(self):
        self.fact_cache = {0: 1}
        self.inv_fact_cache = {0: 1}
        self.prime_factors_map = {}

    def mod_inverse(self, base, mod):
        res = 1
        exp = mod - 2
        while exp > 0:
            if exp % 2 == 1:
                res = res * base % mod
            base = base * base % mod
            exp //= 2
        return res

    def compute_factorial_and_inverse(self, limit):
        if limit in self.fact_cache:
            return
        
        max_computed = max(self.fact_cache.keys())
        for i in range(max_computed + 1, limit + 1):
            self.fact_cache[i] = self.fact_cache[i - 1] * i % MOD_VALUE
            self.inv_fact_cache[i] = self.mod_inverse(self.fact_cache[i], MOD_VALUE)

    def compute_prime_factors_count(self, number):
        if number in self.prime_factors_map:
            return
        
        x = number
        factors = []
        divisor = 2
        while divisor * divisor <= x:
            if x % divisor == 0:
                count = 0
                while x % divisor == 0:
                    count += 1
                    x //= divisor
                factors.append(count)
            divisor += 1
        if x > 1:
            factors.append(1)
        
        self.prime_factors_map[number] = factors

    def combination(self, n, k):
        if k > n or k < 0:
            return 0
        
        self.compute_factorial_and_inverse(n)
        return (self.fact_cache[n] * self.inv_fact_cache[k] % MOD_VALUE * self.inv_fact_cache[n - k] % MOD_VALUE)

    def waysToFillArray(self, queries):
        result = []
        for n, k in queries:
            self.compute_prime_factors_count(k)
            total_ways = 1
            for factor_count in self.prime_factors_map[k]:
                total_ways = total_ways * self.combination(factor_count + n - 1, n - 1) % MOD_VALUE
            result.append(total_ways)
        return result

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