Number of Beautiful Integers in the Range
<p>You are given positive integers <code>low</code>, <code>high</code>, and <code>k</code>.</p>

<p>A number is <strong>beautiful</strong> if it meets both of the following conditions:</p>

<ul>
	<li>The count of even digits in the number is equal to the count of odd digits.</li>
	<li>The number is divisible by <code>k</code>.</li>
</ul>

<p>Return <em>the number of beautiful integers in the range</em> <code>[low, high]</code>.</p>

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

<pre>
<strong>Input:</strong> low = 10, high = 20, k = 3
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are 2 beautiful integers in the given range: [12,18]. 
- 12 is beautiful because it contains 1 odd digit and 1 even digit, and is divisible by k = 3.
- 18 is beautiful because it contains 1 odd digit and 1 even digit, and is divisible by k = 3.
Additionally we can see that:
- 16 is not beautiful because it is not divisible by k = 3.
- 15 is not beautiful because it does not contain equal counts even and odd digits.
It can be shown that there are only 2 beautiful integers in the given range.
</pre>

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

<pre>
<strong>Input:</strong> low = 1, high = 10, k = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> There is 1 beautiful integer in the given range: [10].
- 10 is beautiful because it contains 1 odd digit and 1 even digit, and is divisible by k = 1.
It can be shown that there is only 1 beautiful integer in the given range.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> low = 5, high = 5, k = 2
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are 0 beautiful integers in the given range.
- 5 is not beautiful because it is not divisible by k = 2 and it does not contain equal even and odd digits.
</pre>

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

<ul>
	<li><code>0 &lt; low &lt;= high &lt;= 10<sup>9</sup></code></li>
	<li><code>0 &lt; k &lt;= 20</code></li>
</ul>
Math; Dynamic Programming
function numberOfBeautifulIntegers(low: number, high: number, k: number): number {
    let s = String(high);
    let f: number[][][] = Array(11)
        .fill(0)
        .map(() =>
            Array(21)
                .fill(0)
                .map(() => Array(21).fill(-1)),
        );
    const dfs = (pos: number, mod: number, diff: number, lead: boolean, limit: boolean): number => {
        if (pos >= s.length) {
            return mod == 0 && diff == 10 ? 1 : 0;
        }
        if (!lead && !limit && f[pos][mod][diff] != -1) {
            return f[pos][mod][diff];
        }
        let ans = 0;
        const up = limit ? Number(s[pos]) : 9;
        for (let i = 0; i <= up; ++i) {
            if (i === 0 && lead) {
                ans += dfs(pos + 1, mod, diff, true, limit && i === up);
            } else {
                const nxt = diff + (i % 2 ? 1 : -1);
                ans += dfs(pos + 1, (mod * 10 + i) % k, nxt, false, limit && i === up);
            }
        }
        if (!lead && !limit) {
            f[pos][mod][diff] = ans;
        }
        return ans;
    };
    const a = dfs(0, 0, 10, true, true);
    s = String(low - 1);
    f = Array(11)
        .fill(0)
        .map(() =>
            Array(21)
                .fill(0)
                .map(() => Array(21).fill(-1)),
        );
    const b = dfs(0, 0, 10, true, true);
    return a - b;
}
