<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:
        # Precompute modular values
        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

        # Initialize memoization arrays
        dp = [[-1] * k for _ in range((n + 1) // 2)]
        up = [-1] * ((n + 1) // 2)

        def solve(i, mod):
            if i == (n + 1) // 2:
                return mod == 0
            if dp[i][mod] != -1:
                return dp[i][mod]

            oi = n - i - 1
            for dig in range(9, -1, -1):
                new_mod = (mod + kmod[i][dig] + (kmod[oi][dig] if oi != i else 0)) % k
                if solve(i + 1, new_mod):
                    up[i] = dig
                    break

            dp[i][mod] = up[i] != -1
            return dp[i][mod]

        solve(0, 0)

        # Construct the palindrome
        ans = ['0'] * n
        for i in range((n + 1) // 2):
            ans[i] = ans[n - i - 1] = str(up[i])

        return ''.join(ans)
56656