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#.
What's the output of running the following function using input [30, 20, 10, 100, 33, 12]
?
1def fun(arr: List[int]) -> List[int]:
2 import heapq
3 heapq.heapify(arr)
4 res = []
5 for i in range(3):
6 res.append(heapq.heappop(arr))
7 return res
8
1public static int[] fun(int[] arr) {
2 int[] res = new int[3];
3 PriorityQueue<Integer> heap = new PriorityQueue<>();
4 for (int i = 0; i < arr.length; i++) {
5 heap.add(arr[i]);
6 }
7 for (int i = 0; i < 3; i++) {
8 res[i] = heap.poll();
9 }
10 return res;
11}
12
1class HeapItem {
2 constructor(item, priority = item) {
3 this.item = item;
4 this.priority = priority;
5 }
6}
7
8class MinHeap {
9 constructor() {
10 this.heap = [];
11 }
12
13 push(node) {
14 // insert the new node at the end of the heap array
15 this.heap.push(node);
16 // find the correct position for the new node
17 this.bubble_up();
18 }
19
20 bubble_up() {
21 let index = this.heap.length - 1;
22
23 while (index > 0) {
24 const element = this.heap[index];
25 const parentIndex = Math.floor((index - 1) / 2);
26 const parent = this.heap[parentIndex];
27
28 if (parent.priority <= element.priority) break;
29 // if the parent is bigger than the child then swap the parent and child
30 this.heap[index] = parent;
31 this.heap[parentIndex] = element;
32 index = parentIndex;
33 }
34 }
35
36 pop() {
37 const min = this.heap[0];
38 this.heap[0] = this.heap[this.size() - 1];
39 this.heap.pop();
40 this.bubble_down();
41 return min;
42 }
43
44 bubble_down() {
45 let index = 0;
46 let min = index;
47 const n = this.heap.length;
48
49 while (index < n) {
50 const left = 2 * index + 1;
51 const right = left + 1;
52
53 if (left < n && this.heap[left].priority < this.heap[min].priority) {
54 min = left;
55 }
56 if (right < n && this.heap[right].priority < this.heap[min].priority) {
57 min = right;
58 }
59 if (min === index) break;
60 [this.heap[min], this.heap[index]] = [this.heap[index], this.heap[min]];
61 index = min;
62 }
63 }
64
65 peek() {
66 return this.heap[0];
67 }
68
69 size() {
70 return this.heap.length;
71 }
72}
73
74function fun(arr) {
75 const heap = new MinHeap();
76 for (const x of arr) {
77 heap.push(new HeapItem(x));
78 }
79 const res = [];
80 for (let i = 0; i < 3; i++) {
81 res.push(heap.pop().item);
82 }
83 return res;
84}
85
Solution Implementation
A heap is a ...?
Recommended Readings
Patterns The Shortest Path Algorithm for Coding Interviews The goal of AlgoMonster is to help you get a job in the shortest amount of time possible in a data driven way We compiled datasets of tech interview problems and broke them down by patterns This way we can determine the
Recursion Recursion is one of the most important concepts in computer science Simply speaking recursion is the process of a function calling itself Using a real life analogy imagine a scenario where you invite your friends to lunch https algomonster s3 us east 2 amazonaws com recursion jpg You first
Runtime Overview When learning about algorithms and data structures you'll frequently encounter the term time complexity This concept is fundamental in computer science and offers insights into how long an algorithm takes to complete given a certain input size What is Time Complexity Time complexity represents the amount of time
Got a question? Ask the Monster Assistant anything you don't understand.
Still not clear? Ask in the Forum,  Discord or Submit the part you don't understand to our editors.