<p>You are given two <strong>positive</strong> integers <code>n</code> and <code>k</code>.</p>

<p>An integer <code>x</code> is called <strong>k-palindromic</strong> if:</p>

<ul>
	<li><code>x</code> is a <span data-keyword="palindrome-integer">palindrome</span>.</li>
	<li><code>x</code> is divisible by <code>k</code>.</li>
</ul>

<p>Return the<strong> largest</strong> integer having <code>n</code> digits (as a string) that is <strong>k-palindromic</strong>.</p>

<p><strong>Note</strong> that the integer must <strong>not</strong> have leading zeros.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 3, k = 5</span></p>

<p><strong>Output:</strong> <span class="example-io">&quot;595&quot;</span></p>

<p><strong>Explanation:</strong></p>

<p>595 is the largest k-palindromic integer with 3 digits.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 1, k = 4</span></p>

<p><strong>Output:</strong> <span class="example-io">&quot;8&quot;</span></p>

<p><strong>Explanation:</strong></p>

<p>4 and 8 are the only k-palindromic integers with 1 digit.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 5, k = 6</span></p>

<p><strong>Output:</strong> <span class="example-io">&quot;89898&quot;</span></p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= k &lt;= 9</code></li>
</ul>

class Solution:
    def largestPalindrome(self, n: int, k: int) -> str:
        # memorize + print method
        pow10 = [1 for _ in range(n)]
        for i in range(1, n):
            pow10[i] = (pow10[i-1] * 10) % k

        ans = ["" for _ in range(n)]
        visited = [[False for _ in range(k)] for _ in range((n+1) // 2 + 1)]
        # try to fill one digits by one digits
        # dfs(i, j) means fill i digits with remainder j
        def dfs(i, j):
            # fill all digits to construct a palindrome and remainder == 0
            if i == (n+1) // 2:
                return j == 0
            
            visited[i][j] = True
            # keep filling the rest digits
            for d in range(9, -1, -1):

                # in the middle
                if n % 2 == 1 and i == n // 2:
                    remain = (d*pow10[i] + j) % k
                else:
                    remain = (d*pow10[n-1-i] + d*pow10[i] + j) % k
                if not visited[i+1][remain] and dfs(i+1, remain):
                    ans[i] = str(d)
                    ans[n-1-i] = str(d)
                    return True

            return False

        dfs(0, 0)
        
        return "".join(ans)


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