Concatenated Divisibility - Problem
You are given an array of positive integers nums and a positive integer k.
A permutation of nums is said to form a divisible concatenation if, when you concatenate the decimal representations of the numbers in the order specified by the permutation, the resulting number is divisible by k.
Return the lexicographically smallest permutation (when considered as a list of integers) that forms a divisible concatenation. If no such permutation exists, return an empty list.
Example: For nums = [232, 124, 456] and k = 4, the permutation [124, 232, 456] creates the number 124232456, which we need to check if it's divisible by 4.
Input & Output
example_1.py — Basic Case
$
Input:
nums = [232, 124, 456], k = 4
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Output:
[124, 232, 456]
💡 Note:
The concatenation '124232456' is divisible by 4. This is the lexicographically smallest valid permutation.
example_2.py — No Solution
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Input:
nums = [11, 13, 17], k = 7
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Output:
[]
💡 Note:
No permutation of these numbers creates a concatenation divisible by 7, so we return an empty array.
example_3.py — Single Element
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Input:
nums = [1], k = 1
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Output:
[1]
💡 Note:
Single element case - the number 1 is divisible by 1, so we return [1].
Visualization
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Understanding the Visualization
1
Problem Setup
We have number tiles [232, 124, 456] and need divisibility by k=4
2
Try Arrangements
Test different arrangements: 232124456, 124232456, etc.
3
Check Divisibility
Use modular arithmetic to check if concatenated result is divisible by k
4
Find Smallest
Among valid arrangements, return the lexicographically smallest
Key Takeaway
🎯 Key Insight: Use dynamic programming with bitmasks to efficiently explore permutations while using modular arithmetic to avoid integer overflow when checking divisibility.
Time & Space Complexity
Time Complexity
O(n × 2^n × k)
n numbers, 2^n possible masks, k possible remainders
⚠ Quadratic Growth
Space Complexity
O(2^n × k)
DP table storing states for each mask and remainder combination
⚠ Quadratic Space
Constraints
- 1 ≤ nums.length ≤ 8
- 1 ≤ nums[i] ≤ 108
- 1 ≤ k ≤ 400
- All integers in nums are unique
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Explanation
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