<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 numberOfBeautifulIntegers(self, low: int, high: int, k: int) -> int:
        @cache
        def go(thresh, deg, odd, rem):
            if deg == - 1:
                return int(odd == 0 and rem == 0)
            res = 0
            for i in range(10):
                if thresh is inf or i <= int(str(thresh)[- 1 - deg]):
                    res += go(
                        thresh if thresh is inf or i == int(str(thresh)[- 1 - deg]) else inf,
                        deg - 1,
                        odd + (1 if i % 2 == 1 else - 1),
                        (rem + pow(10, deg, k) * i) % k
                        )
            return res

        def full_count(thresh):
            # need at least 2 digits
            if thresh < 10:
                return 0
            res = 0
            for start_deg in range(1, len(str(thresh)), 2):
                for i in range(1, 10):
                    if 10 ** start_deg * i <= thresh:
                        res += go(
                            thresh if start_deg + 1 == len(str(thresh)) and str(i) == str(thresh)[0] else inf, 
                            start_deg - 1, 
                            (i % 2) * 2 - 1, 
                            pow(10, start_deg, k) * i % k
                            )
            return res

        return full_count(high) - full_count(low - 1)
        
17100