<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>

pos_mod_k={}
for k in range(1,10):
    pos_mod=[[0]*(10**5+1) for _ in range(10)]
    for d in range(10):
        pos_mod[d][0]=d%k
    for d in range(10):
        for i in range(1,10**5+1):
            pos_mod[d][i]=(pos_mod[d][i-1]*10)%k
    pos_mod_k[k]=pos_mod
class Solution:
    def largestPalindrome(self, n: int, k: int) -> str:
        choice=[None]*n
        half_len=(n+1)//2
        @cache
        def dp(i, mod):
            if i == half_len:
                if mod == 0:
                    return True
                return False
            digit_pos = set([i, n - i - 1])
            for d in range(9, -1, -1):
                cur_mod = 0
                for pos in digit_pos:
                    choice[pos] = d
                    cur_mod += pos_mod_k[k][d][pos]
                if dp(i + 1, (cur_mod + mod) % k):
                    return True
            return False
        dp(0, 0)
        return ''.join(map(str,choice))
221808