<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 get_number_of_beautiful_integers(self, high_str, k, pos, started, under_high, even_digit_count, total_digit_count, remainder, memoization):
        if pos == len(high_str):
            if even_digit_count == total_digit_count - even_digit_count and remainder == 0:
                return 1
            else:
                return 0
        
        if (pos, under_high, even_digit_count, total_digit_count, remainder) in memoization:
            return memoization[(pos, under_high, even_digit_count, total_digit_count, remainder)]
        
        res = 0
        for i in range(10):
            current_started = started or i > 0
            high_digit = int(high_str[pos])
            current_under_high = under_high or i < high_digit

            if not current_under_high and i > high_digit:
                break
            
            current_total_digit_count = total_digit_count
            current_even_digit_count = even_digit_count
            current_remainder = remainder
            if current_started:
                current_total_digit_count += 1
                if i % 2 == 0:
                    current_even_digit_count += 1
                current_remainder = (remainder * 10 + i) % k
            res += self.get_number_of_beautiful_integers(high_str, k, pos+1, current_started, current_under_high,
                                                         current_even_digit_count, current_total_digit_count, current_remainder, memoization)
        memoization[(pos, under_high, even_digit_count, total_digit_count, remainder)] = res
        return res


    def numberOfBeautifulIntegers(self, low: int, high: int, k: int) -> int:
        return self.get_number_of_beautiful_integers(str(high), k, 0, False, False, 0, 0, 0, {}) - self.get_number_of_beautiful_integers(str(low-1), k, 0, False, False, 0, 0, 0, {})
17000