Count Ways to Make Array With Product
<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>
Array; Math; Dynamic Programming; Combinatorics; Number Theory
class Solution {
    private static final int N = 10020;
    private static final int MOD = (int) 1e9 + 7;
    private static final long[] F = new long[N];
    private static final long[] G = new long[N];
    private static final List<Integer>[] P = new List[N];

    static {
        F[0] = 1;
        G[0] = 1;
        Arrays.setAll(P, k -> new ArrayList<>());
        for (int i = 1; i < N; ++i) {
            F[i] = F[i - 1] * i % MOD;
            G[i] = qmi(F[i], MOD - 2, MOD);
            int x = i;
            for (int j = 2; j <= x / j; ++j) {
                if (x % j == 0) {
                    int cnt = 0;
                    while (x % j == 0) {
                        ++cnt;
                        x /= j;
                    }
                    P[i].add(cnt);
                }
            }
            if (x > 1) {
                P[i].add(1);
            }
        }
    }

    public static long qmi(long a, long k, long p) {
        long res = 1;
        while (k != 0) {
            if ((k & 1) == 1) {
                res = res * a % p;
            }
            k >>= 1;
            a = a * a % p;
        }
        return res;
    }

    public static long comb(int n, int k) {
        return (F[n] * G[k] % MOD) * G[n - k] % MOD;
    }

    public int[] waysToFillArray(int[][] queries) {
        int m = queries.length;
        int[] ans = new int[m];
        for (int i = 0; i < m; ++i) {
            int n = queries[i][0], k = queries[i][1];
            long t = 1;
            for (int x : P[k]) {
                t = t * comb(x + n - 1, n - 1) % MOD;
            }
            ans[i] = (int) t;
        }
        return ans;
    }
}