<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:
        kmod = [[0] * 10 for _ in range(n)]
        num = 1
        for i in range(n):
            for j in range(1, 10):
                kmod[i][j] = (num * j) % k
            num = (num * 10) % k
            
        @cache
        def dp(index, remainder):
            if n % 2 == 1 and index == n // 2:
                for i in range(9, 0, -1):
                    if (kmod[index][i] + remainder) % k == 0:
                        return [str(i)]
            elif index >= n // 2:
                return -1 if remainder != 0 else [""]

            for i in range(9, 0, -1):
                newRemainder = (kmod[n - index -1][i] + kmod[index][i] + remainder) % k
                if dp(index + 1, newRemainder) != -1:
                    result = dp(index + 1, newRemainder)
                    result.append(str(i))
                    return result
            return -1
        
        result = dp(0, 0)
        result.reverse()
        for i in range(n // 2):
            result.append(result[n // 2 - i - 1])
        return "".join(result)

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