2183. Count Array Pairs Divisible by K
Problem Description
Given an array nums
and an integer k
, we have to find the number of pairs (i,j)
such that the product of nums[i]
and nums[j]
is divisible by k
. Note that k
can be as large as 1e5.
Here's a high-level approach to solving this problem:
-
Calculate the greatest common divisor (gcd) of each value in
nums
withk
. -
Then, for each pair
(nums[i], nums[j])
, check if their gcd values multiplied together are divisible byk
. If yes, add them to the count.
Let's walk through an example to better understand the approach.
Example
Suppose nums = [1, 2, 3, 4, 5] and k = 6. We first find the gcd values for each element in the array nums
with k.
1 x 6 = 6, gcd(1, 6) = 1 2 x 3 = 6, gcd(2, 6) = 2 3 x 2 = 6, gcd(3, 6) = 3 4 x 1.5 = 6, gcd(4, 6) = 2 5 x 1.2 = 6, gcd(5, 6) = 1
Now the gcd values are [1, 2, 3, 2, 1].
We go through each pair of gcd values and check if their product is divisible by k
. In our example:
(1, 2) - No (1, 3) - No (1, 2) - No (1, 1) - No (2, 3) - Yes (2, 2) - No (2, 1) - No (3, 2) - Yes (3, 1) - No (2, 1) - No
We find two pairs that are divisible by k
: (2, 3) and (3, 2). So the final answer is 2.
Solution
Python
1 2python 3from math import gcd 4from typing import List 5 6 7class Solution: 8 def countPairs(self, nums: List[int], k: int) -> int: 9 ans = 0 10 gcds = {} 11 12 for num in nums: 13 gcd_num = gcd(num, k) 14 for gcd_val, count in gcds.items(): 15 if gcd_num * gcd_val % k == 0: 16 ans += count 17 gcds[gcd_num] = gcds.get(gcd_num, 0) + 1 18 19 return ans
Java
1
2java
3import java.util.HashMap;
4import java.util.Map;
5
6class Solution {
7 public long countPairs(int[] nums, int k) {
8 long ans = 0;
9 Map<Integer, Integer> gcds = new HashMap<>();
10
11 for (int num : nums) {
12 int gcdNum = gcd(num, k);
13 for (Map.Entry<Integer, Integer> entry : gcds.entrySet()) {
14 if (gcdNum * entry.getKey() % k == 0) {
15 ans += entry.getValue();
16 }
17 }
18 gcds.put(gcdNum, gcds.getOrDefault(gcdNum, 0) + 1);
19 }
20
21 return ans;
22 }
23
24 private int gcd(int a, int b) {
25 if (b == 0) {
26 return a;
27 } else {
28 return gcd(b, a % b);
29 }
30 }
31}
JavaScript
1
2javascript
3class Solution {
4 countPairs(nums, k) {
5 let ans = 0;
6 const gcds = new Map();
7
8 for (const num of nums) {
9 const gcdNum = this.gcd(num, k);
10 for (const [gcdVal, count] of gcds.entries()) {
11 if (gcdNum * gcdVal % k === 0) {
12 ans += count;
13 }
14 }
15 gcds.set(gcdNum, (gcds.get(gcdNum) || 0) + 1);
16 }
17
18 return ans;
19 }
20
21 gcd(a, b) {
22 if (b === 0) {
23 return a;
24 } else {
25 return this.gcd(b, a % b);
26 }
27 }
28}
C++
1
2cpp
3#include <unordered_map>
4
5class Solution {
6public:
7 long long countPairs(vector<int>& nums, int k) {
8 long ans = 0;
9 unordered_map<int, int> gcds;
10
11 for (const int num : nums) {
12 const int gcdNum = gcd(num, k);
13 for (const auto& [gcdVal, count] : gcds) {
14 if (gcdNum * gcdVal % k == 0) {
15 ans += count;
16 }
17 }
18 ++gcds[gcdNum];
19 }
20
21 return ans;
22 }
23
24private:
25 int gcd(int a, int b) {
26 if (b == 0) {
27 return a;
28 } else {
29 return gcd(b, a % b);
30 }
31 }
32};
C#
1
2csharp
3using System.Collections.Generic;
4
5public class Solution {
6 public long CountPairs(int[] nums, int k) {
7 long ans = 0;
8 Dictionary<int, int> gcds = new Dictionary<int, int>();
9
10 foreach (int num in nums) {
11 int gcdNum = GCD(num, k);
12 foreach (KeyValuePair<int, int> entry in gcds) {
13 if (gcdNum * entry.Key % k == 0) {
14 ans += entry.Value;
15 }
16 }
17 if (gcds.ContainsKey(gcdNum)) {
18 gcds[gcdNum]++;
19 } else {
20 gcds[gcdNum] = 1;
21 }
22 }
23
24 return ans;
25 }
26
27 private int GCD(int a, int b) {
28 if (b == 0) {
29 return a;
30 } else {
31 return GCD(b, a % b);
32 }
33 }
34}
In conclusion, the above solutions follow the high-level approach explained earlier and find the count of pairs such that their product is divisible by k
. Each solution utilizes the greatest common divisor (gcd) function to calculate the gcd values, iterate through each pair of gcd values, and check if their product is divisible by k
. The implementation is provided in Python, Java, JavaScript, C++, and C#.
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