<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:
        def helper(num: tuple) -> int:
            @cache
            def dp(pos: int, tight: bool, has_first: bool, even_num: int, cur_num: int) -> int:
                if pos == len(num):
                    if even_num == 0 and cur_num == 0: return 1
                    else: return 0
                res = 0
                bound = num[pos] if tight else 10
                for i in range(bound + 1):
                    if i == 10: continue
                    new_tight = i == bound
                    new_has_first = has_first
                    if i == 0 and not has_first: new_even_num = 0
                    else:
                        new_has_first = True
                        if i % 2 == 0: new_even_num = even_num + 1
                        else: new_even_num = even_num - 1
                    res += dp(
                              pos + 1, 
                              new_tight, 
                              new_has_first, 
                              new_even_num, 
                              (cur_num * 10 + i) % k
                              )
                return res
            return dp(0, True, False, 0, 0)
        
        return helper(tuple(map(int, str(high)))) - helper(tuple(map(int, str(low - 1))))
18000