<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, start: int, end: int, div: int) -> int:
        
        # Cached function to compute beautiful numbers from 0 to s.
        @cache
        def recursive(s, j=0, count=0, withinLimit=True, remainder=0, isZero=True):
            if j == len(s): return 1 if count == 0 and remainder == 0 else 0
            total, max_digit = 0, int(s[j]) if withinLimit else 9
            for digit in range(max_digit + 1):
                isCurrentLimit, isCurrentZero = withinLimit and digit == int(s[j]), isZero and digit == 0
                # Process based on whether the current number has a leading zero.
                if isCurrentZero: total += recursive(s, j + 1, count, isCurrentLimit, remainder, isCurrentZero)
                else: total += recursive(s, j + 1, count + 1 if digit % 2 == 0 else count - 1, isCurrentLimit, (remainder * 10 + digit) % div, isCurrentZero)
            return total

        # Return the difference to get count of beautiful integers in the range.
        return recursive(str(end)) - recursive(str(start - 1))
18600