<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:
        memo = {}
        digits = []
        
        def solve(pos, odd, even, rem, flag, lead_zero):
            if odd > len(digits) or even > len(digits):
                return 0
            if pos < 0:
                return int(odd == even and rem == 0)

            inputs = (pos, odd, even, rem, flag, lead_zero)
            if inputs in memo:
                return memo[inputs]

            answer = 0
            for digit in range(digits[pos] + 1 if flag else 10):
                new_odd = odd
                new_even = even
                if not lead_zero or digit > 0:
                    if digit % 2 == 1: 
                        new_odd += 1
                    else: 
                        new_even += 1
                new_rem = (rem * 10 + digit) % k
                new_flag = flag and (digit == digits[pos])
                new_lead_zero = lead_zero and (digit == 0)
                answer += solve(pos-1, new_odd, new_even, new_rem, new_flag, new_lead_zero)

            memo[inputs] = answer
            return answer

        def process(n):
            memo.clear()
            digits.clear()

            while n > 0:
                digits.append(n % 10)
                n //= 10

            return solve(len(digits)-1, 0, 0, 0, True, True)

        return process(high) - process(low-1)
17800