Find the Count of Good Integers

Problem Description

You are given two positive integers n and k. An integer x is called k-palindromic if it is a palindrome and is divisible by k. An integer is called good if its digits can be rearranged to form a k-palindromic integer. Your task is to return the count of good integers containing n digits.

Key Insights

• A k-palindromic integer must be a palindrome and divisible by k.
• The digits of a good integer can be rearranged, so the frequency of each digit matters.
• Leading zeros are not allowed, which affects the count of valid configurations.
• The problem involves combinatorial counting of digit arrangements while ensuring palindromic properties.

Space and Time Complexity

Time Complexity: O(n^2) - Due to combinatorial checks and arrangements for digits. Space Complexity: O(1) - Only a fixed number of integer variables are used.

Solution

To solve the problem, we will:

1. Count the frequency of each digit in the range from 1 to 9 (since leading zeros are not allowed).
2. Utilize combinatorial mathematics to determine valid arrangements of digits that can form a palindrome.
3. Check for divisibility by k for the resulting palindromic integers.

We will use a hash table to store the frequency of digits and a mathematical approach to count palindromic arrangements.

Code Solutions