<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>

class Solution:
    def count(self, num, i, n, tight, nzseen, mod, k, diff, cache):
        if i >= n:
            return int(mod == 0 and diff == 0)
        key = (i, tight, nzseen, mod, diff)
        if key in cache:
            return cache[key]
        maxd = 9
        if tight:
            maxd = int(num[i])
        res = 0
        for d in range(maxd + 1):
            newnzseen = nzseen or d != 0
            newdiff = diff
            if newnzseen:
                if d & 1:
                    newdiff += 1
                else:
                    newdiff -= 1
            res += self.count(num, i + 1, n, tight and d == maxd, newnzseen, (10 * mod + d) % k, k, newdiff, cache)
        cache[key] = res
        return res
    
    def numberOfBeautifulIntegers(self, low: int, high: int, k: int) -> int:
        return self.count(str(high), 0, len(str(high)), True, False, 0, k, 0, {}) - self.count(str(low - 1), 0, len(str(low - 1)), True, False, 0, k, 0, {})
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