<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 ddp(num):
            if num <= 9:
                return 0
            # isnum, islim, diff, mod
            d = defaultdict(int)
            s = str(num)
            lens = len(s)

            d[(0, 0, 0, 0)] = 1
            s0 = int(s[0])
            for i in range(1, s0):
                d[(1, 0, mem[i], i % k)] += 1
            d[(1, 1, mem[s0], s0 % k)] = 1

            for i in range(1, lens):
                tmp = defaultdict(int)
                s0 = int(s[i])
                for (isnum, islim, diff, mod), val in d.items():

                    if isnum == 0:
                        tmp[(0, 0, 0, 0)] += 1
                        for i in range(1, 10):
                            tmp[(1, 0, mem[i], i % k)] += 1

                    else:
                        if islim == 1:
                            for i in range(s0):
                                tmp[(1, 0, diff + mem[i], (mod * 10 + i) % k)] += val
                            tmp[(1, 1, diff + mem[s0], (mod * 10 + s0) % k)] += val

                        else:
                            for i in range(10):
                                tmp[(1, 0, diff + mem[i], (mod * 10 + i) % k)] += val

                d = tmp

            res = 0
            for (isnum, islim, diff, mod), val in d.items():
                if isnum == 1 and diff == 0 and mod == 0:
                    res += val

            return res

        # even -1, odd +1
        mem = [-1, 1] * 5

        return ddp(high) - ddp(low - 1)

16700