Find all pairs (a, b) in an array such that a % b = k
Last Updated :
24 Oct, 2024
Given an array with distinct elements, the task is to find the pairs in the array such that a % b = k, where k is a given integer. You may assume that a and b are in small range
Examples :
Input : arr[] = {2, 3, 5, 4, 7}
k = 3
Output : (7, 4), (3, 4), (3, 5), (3, 7)
7 % 4 = 3
3 % 4 = 3
3 % 5 = 3
3 % 7 = 3
Input: arr[] = [1, 2], k = 3
Output: 0
Explanation: No pairs give remainder 3.
Naive Approach - O(n^2) Time and O(1) Space
We consider all pairs one by one and check their modulo is equal to k or not. If equals to k, then print that pair.
C++
// C++ implementation to find such pairs
#include <bits/stdc++.h>
using namespace std;
// Function to find pair such that (a % b = k)
bool printPairs(int arr[], int n, int k)
{
bool isPairFound = true;
// Consider each and every pair
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
// Print if their modulo equals to k
if (i != j && arr[i] % arr[j] == k) {
cout << "(" << arr[i] << ", "
<< arr[j] << ")"
<< " ";
isPairFound = true;
}
}
}
return isPairFound;
}
// Driver program
int main()
{
int arr[] = { 2, 3, 5, 4, 7 };
int n = sizeof(arr) / sizeof(arr[0]);
int k = 3;
if (printPairs(arr, n, k) == false)
cout << "No such pair exists";
return 0;
}
// Java implementation to find such pairs
class Test {
// method to find pair such that (a % b = k)
static boolean printPairs(int arr[], int n, int k)
{
boolean isPairFound = true;
// Consider each and every pair
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
// Print if their modulo equals to k
if (i != j && arr[i] % arr[j] == k) {
System.out.print("(" + arr[i] + ", " + arr[j] + ")"
+ " ");
isPairFound = true;
}
}
}
return isPairFound;
}
// Driver method
public static void main(String args[])
{
int arr[] = { 2, 3, 5, 4, 7 };
int k = 3;
if (printPairs(arr, arr.length, k) == false)
System.out.println("No such pair exists");
}
}
# Python3 implementation to find such pairs
# Function to find pair such that (a % b = k)
def printPairs(arr, n, k):
isPairFound = True
# Consider each and every pair
for i in range(0, n):
for j in range(0, n):
# Print if their modulo equals to k
if (i != j and arr[i] % arr[j] == k):
print("(", arr[i], ", ", arr[j], ")",
sep = "", end = " ")
isPairFound = True
return isPairFound
# Driver Code
arr = [2, 3, 5, 4, 7]
n = len(arr)
k = 3
if (printPairs(arr, n, k) == False):
print("No such pair exists")
#
// C# implementation to find such pair
using System;
public class GFG {
// method to find pair such that (a % b = k)
static bool printPairs(int[] arr, int n, int k)
{
bool isPairFound = true;
// Consider each and every pair
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
{
// Print if their modulo equals to k
if (i != j && arr[i] % arr[j] == k)
{
Console.Write("(" + arr[i] + ", "
+ arr[j] + ")" + " ");
isPairFound = true;
}
}
}
return isPairFound;
}
// Driver method
public static void Main()
{
int[] arr = { 2, 3, 5, 4, 7 };
int k = 3;
if (printPairs(arr, arr.Length, k) == false)
Console.WriteLine("No such pair exists");
}
}
// This code is contributed by Sam007
<script>
// Javascript implementation to find such pair
// method to find pair such that (a % b = k)
function printPairs(arr, n, k)
{
let isPairFound = true;
// Consider each and every pair
for (let i = 0; i < n; i++) {
for (let j = 0; j < n; j++)
{
// Print if their modulo equals to k
if (i != j && arr[i] % arr[j] == k)
{
document.write("(" + arr[i] + ", " + arr[j] + ")" + " ");
isPairFound = true;
}
}
}
return isPairFound;
}
let arr = [ 2, 3, 5, 4, 7 ];
let k = 3;
if (printPairs(arr, arr.length, k) == false)
document.write("No such pair exists");
</script>
<?php
// PHP implementation to
// find such pairs
// Function to find pair
// such that (a % b = k)
function printPairs($arr, $n, $k)
{
$isPairFound = true;
// Consider each and every pair
for ($i = 0; $i < $n; $i++)
{
for ( $j = 0; $j < $n; $j++)
{
// Print if their modulo
// equals to k
if ($i != $j && $arr[$i] %
$arr[$j] == $k)
{
echo "(" , $arr[$i] , ", ",
$arr[$j] , ")", " ";
$isPairFound = true;
}
}
}
return $isPairFound;
}
// Driver Code
$arr = array(2, 3, 5, 4, 7);
$n = sizeof($arr);
$k = 3;
if (printPairs($arr, $n, $k) == false)
echo "No such pair exists";
// This code is contributed by ajit
?>
Output(3, 5) (3, 4) (3, 7) (7, 4)
Efficient Solution for Small Values and Large Arrays - O(n* sqrt(max))) Time and O(n) Space
The idea is based on below observations :
- If k itself is present in arr[], then k forms a pair with all elements arr[i] where k < arr[i]. For all such arr[i], we have k % arr[i] = k.
- For all elements greater than or equal to k, we use the following fact.
If arr[i] % arr[j] = k,
==> arr[i] = x * arr[j] + k
==> (arr[i] - k) = x * arr[j]
We find all divisors of (arr[i] - k)
and see if they are present in arr[].
To quickly check if an element is present in the array, we use hashing.
C++
// C++ program to find all pairs such that
// a % b = k.
#include <bits/stdc++.h>
using namespace std;
// Utility function to find the divisors of
// n and store in vector v[]
vector<int> findDivisors(int n)
{
vector<int> v;
// Vector is used to store the divisors
for (int i = 1; i <= sqrt(n); i++) {
if (n % i == 0) {
// If n is a square number, push
// only one occurrence
if (n / i == i)
v.push_back(i);
else {
v.push_back(i);
v.push_back(n / i);
}
}
}
return v;
}
// Function to find pairs such that (a%b = k)
bool printPairs(int arr[], int n, int k)
{
// Store all the elements in the map
// to use map as hash for finding elements
// in O(1) time.
unordered_map<int, bool> occ;
for (int i = 0; i < n; i++)
occ[arr[i]] = true;
bool isPairFound = false;
for (int i = 0; i < n; i++) {
// Print all the pairs with (a, b) as
// (k, numbers greater than k) as
// k % (num (> k)) = k i.e. 2%4 = 2
if (occ[k] && k < arr[i]) {
cout << "(" << k << ", " << arr[i] << ") ";
isPairFound = true;
}
// Now check for the current element as 'a'
// how many b exists such that a%b = k
if (arr[i] >= k) {
// find all the divisors of (arr[i]-k)
vector<int> v = findDivisors(arr[i] - k);
// Check for each divisor i.e. arr[i] % b = k
// or not, if yes then print that pair.
for (int j = 0; j < v.size(); j++) {
if (arr[i] % v[j] == k && arr[i] != v[j] && occ[v[j]]) {
cout << "(" << arr[i] << ", "
<< v[j] << ") ";
isPairFound = true;
}
}
// Clear vector
v.clear();
}
}
return isPairFound;
}
// Driver program
int main()
{
int arr[] = { 3, 1, 2, 5, 4 };
int n = sizeof(arr) / sizeof(arr[0]);
int k = 2;
if (printPairs(arr, n, k) == false)
cout << "No such pair exists";
return 0;
}
// Java program to find all pairs such that
// a % b = k.
import java.util.HashMap;
import java.util.Vector;
class Test {
// Utility method to find the divisors of
// n and store in vector v[]
static Vector<Integer> findDivisors(int n)
{
Vector<Integer> v = new Vector<>();
// Vector is used to store the divisors
for (int i = 1; i <= Math.sqrt(n); i++) {
if (n % i == 0) {
// If n is a square number, push
// only one occurrence
if (n / i == i)
v.add(i);
else {
v.add(i);
v.add(n / i);
}
}
}
return v;
}
// method to find pairs such that (a%b = k)
static boolean printPairs(int arr[], int n, int k)
{
// Store all the elements in the map
// to use map as hash for finding elements
// in O(1) time.
HashMap<Integer, Boolean> occ = new HashMap<>();
for (int i = 0; i < n; i++)
occ.put(arr[i], true);
boolean isPairFound = false;
for (int i = 0; i < n; i++) {
// Print all the pairs with (a, b) as
// (k, numbers greater than k) as
// k % (num (> k)) = k i.e. 2%4 = 2
if (occ.get(k) && k < arr[i]) {
System.out.print("(" + k + ", " + arr[i] + ") ");
isPairFound = true;
}
// Now check for the current element as 'a'
// how many b exists such that a%b = k
if (arr[i] >= k) {
// find all the divisors of (arr[i]-k)
Vector<Integer> v = findDivisors(arr[i] - k);
// Check for each divisor i.e. arr[i] % b = k
// or not, if yes then print that pair.
for (int j = 0; j < v.size(); j++) {
if (arr[i] % v.get(j) == k && arr[i] != v.get(j) && occ.get(v.get(j))) {
System.out.print("(" + arr[i] + ", "
+ v.get(j) + ") ");
isPairFound = true;
}
}
// Clear vector
v.clear();
}
}
return isPairFound;
}
// Driver method
public static void main(String args[])
{
int arr[] = { 3, 1, 2, 5, 4 };
int k = 2;
if (printPairs(arr, arr.length, k) == false)
System.out.println("No such pair exists");
}
}
# Python3 program to find all pairs
# such that a % b = k.
# Utility function to find the divisors
# of n and store in vector v[]
import math as mt
def findDivisors(n):
v = []
# Vector is used to store the divisors
for i in range(1, mt.floor(n**(.5)) + 1):
if (n % i == 0):
# If n is a square number, push
# only one occurrence
if (n / i == i):
v.append(i)
else:
v.append(i)
v.append(n // i)
return v
# Function to find pairs such that (a%b = k)
def printPairs(arr, n, k):
# Store all the elements in the map
# to use map as hash for finding elements
# in O(1) time.
occ = dict()
for i in range(n):
occ[arr[i]] = True
isPairFound = False
for i in range(n):
# Print all the pairs with (a, b) as
# (k, numbers greater than k) as
# k % (num (> k)) = k i.e. 2%4 = 2
if (occ[k] and k < arr[i]):
print("(", k, ",", arr[i], ")", end = " ")
isPairFound = True
# Now check for the current element as 'a'
# how many b exists such that a%b = k
if (arr[i] >= k):
# find all the divisors of (arr[i]-k)
v = findDivisors(arr[i] - k)
# Check for each divisor i.e. arr[i] % b = k
# or not, if yes then print that pair.
for j in range(len(v)):
if (arr[i] % v[j] == k and
arr[i] != v[j] and
occ[v[j]]):
print("(", arr[i], ",", v[j],
")", end = " ")
isPairFound = True
return isPairFound
# Driver Code
arr = [3, 1, 2, 5, 4]
n = len(arr)
k = 2
if (printPairs(arr, n, k) == False):
print("No such pair exists")
# This code is contributed by mohit kumar
// C# program to find all pairs
// such that a % b = k.
using System;
using System.Collections.Generic;
class GFG
{
// Utility method to find the divisors
// of n and store in vector v[]
public static List<int> findDivisors(int n)
{
List<int> v = new List<int>();
// Vector is used to store
// the divisors
for (int i = 1;
i <= Math.Sqrt(n); i++)
{
if (n % i == 0)
{
// If n is a square number,
// push only one occurrence
if (n / i == i)
{
v.Add(i);
}
else
{
v.Add(i);
v.Add(n / i);
}
}
}
return v;
}
// method to find pairs such
// that (a%b = k)
public static bool printPairs(int[] arr,
int n, int k)
{
// Store all the elements in the
// map to use map as hash for
// finding elements in O(1) time.
Dictionary<int,
bool> occ = new Dictionary<int,
bool>();
for (int i = 0; i < n; i++)
{
occ[arr[i]] = true;
}
bool isPairFound = false;
for (int i = 0; i < n; i++)
{
// Print all the pairs with (a, b) as
// (k, numbers greater than k) as
// k % (num (> k)) = k i.e. 2%4 = 2
if (occ[k] && k < arr[i])
{
Console.Write("(" + k + ", " +
arr[i] + ") ");
isPairFound = true;
}
// Now check for the current element
// as 'a' how many b exists such that
// a%b = k
if (arr[i] >= k)
{
// find all the divisors of (arr[i]-k)
List<int> v = findDivisors(arr[i] - k);
// Check for each divisor i.e.
// arr[i] % b = k or not, if
// yes then print that pair.
for (int j = 0; j < v.Count; j++)
{
if (arr[i] % v[j] == k &&
arr[i] != v[j] && occ[v[j]])
{
Console.Write("(" + arr[i] +
", " + v[j] + ") ");
isPairFound = true;
}
}
// Clear vector
v.Clear();
}
}
return isPairFound;
}
// Driver Code
public static void Main(string[] args)
{
int[] arr = new int[] {3, 1, 2, 5, 4};
int k = 2;
if (printPairs(arr, arr.Length, k) == false)
{
Console.WriteLine("No such pair exists");
}
}
}
// This code is contributed by Shrikant13
<script>
// JavaScript program to find all pairs such that
// a % b = k.
// Utility method to find the divisors of
// n and store in vector v[]
function findDivisors(n)
{
let v = [];
// Vector is used to store the divisors
for (let i = 1; i <= Math.sqrt(n); i++)
{
if (n % i == 0) {
// If n is a square number, push
// only one occurrence
if (n / i == i)
v.push(i);
else {
v.push(i);
v.push(Math.floor(n / i));
}
}
}
return v;
}
// method to find pairs such that (a%b = k)
function printPairs(arr,n,k)
{
// Store all the elements in the map
// to use map as hash for finding elements
// in O(1) time.
let occ = new Map();
for (let i = 0; i < n; i++)
occ.set(arr[i], true);
let isPairFound = false;
for (let i = 0; i < n; i++) {
// Print all the pairs with (a, b) as
// (k, numbers greater than k) as
// k % (num (> k)) = k i.e. 2%4 = 2
if (occ.get(k) && k < arr[i]) {
document.write("(" + k + ", " +
arr[i] + ") ");
isPairFound = true;
}
// Now check for the current element as 'a'
// how many b exists such that a%b = k
if (arr[i] >= k) {
// find all the divisors of (arr[i]-k)
let v = findDivisors(arr[i] - k);
// Check for each divisor
// i.e. arr[i] % b = k
// or not, if yes then
// print that pair.
for (let j = 0; j < v.length; j++)
{
if (arr[i] % v[j] == k &&
arr[i] != v[j] &&
occ.get(v[j]))
{
document.write("(" + arr[i] + ", "
+ v[j] + ") ");
isPairFound = true;
}
}
// Clear vector
v=[];
}
}
return isPairFound;
}
// Driver method
let arr=[3, 1, 2, 5, 4 ];
let k = 2;
if (printPairs(arr, arr.length, k) == false)
document.write("No such pair exists");
// This code is contributed by unknown2108
</script>
Output(2, 3) (2, 5) (5, 3) (2, 4)
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