Reach the numbers by making jumps of two given lengths

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Given integers k, d1, d2 and an integer array arr[]. Starting from number k you are allowed to make jumps of size d1 and d2 i.e. all the possible moves from k are:

k + d1 and k – d1
k + d2 and k – d2

The task is to find which of the numbers from the array are reachable from k making any number of jumps and any given jump is either of size d1 or d2. For every number print 1 if its reachable else print 0.

Examples:

Input: k = 10, d1 = 4, d2 = 6, arr[] = {10, 15, 20}
Output: 1 0 1
10 can be reached from k with no extra move.
20 can be reached with k + d1 + d2 = 10 + 4 + 6 = 20

Input: k = 8, d1 = 3, d2 = 2, arr[] = {9, 4}
Output: 1 1

Approach:

Any number x that is reachable from k with jumps d1 or d2 will be of the form x = k + (i * d1) + (j * d2) where i and j are integers.
Let the GCD of d1 and d2 be gcd. Since, gcd divides both d1 and d2. Therefore we can write d1 = m1 * gcd and d2 = m2 * gcd where m1 and m2 are integers
And x = k + gcd * (i * m1 + j * m2) = k + M * gcd.
So, any number x that is reachable from k should satisfy (x – k) % gcd = 0.

Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <algorithm>
#include <iostream>
using namespace std;
 
// Function that returns the vector containing the
// result for the reachability of the required numbers
void reachTheNums(int nums[], int k, int d1, int d2, int n)
{
    int i, ans[n] = { 0 };
    int gcd = __gcd(d1, d2);
 
    for (i = 0; i < n; i++) {
        int x = nums[i] - k;
 
        // If distance x is coverable
        if (x % gcd == 0)
            ans[i] = 1;
        else
            ans[i] = 0;
    }
 
    for (i = 0; i < n; i++)
        cout << ans[i] << " ";
}
 
// Driver code
int main()
{
    // Numbers to be checked for reachability
    int nums[] = { 9, 4 };
    int n = sizeof(nums) / sizeof(nums[0]);
    // Starting number K
    int k = 8;
 
    // Sizes of jumps d1 and d2
    int d1 = 3, d2 = 2;
 
    reachTheNums(nums, k, d1, d2, n);
 
    return 0;
}

Java

// Java implementation of the approach
 
import java.io.*;
class GFG {
// Recursive function to return gcd of a and b 
    static int __gcd(int a, int b) 
    { 
        // Everything divides 0  
        if (a == 0) 
          return b; 
        if (b == 0) 
          return a; 
        
        // base case 
        if (a == b) 
            return a; 
        
        // a is greater 
        if (a > b) 
            return __gcd(a-b, b); 
        return __gcd(a, b-a); 
    } 
 
 
     
 
// Function that returns the vector containing the
// result for the reachability of the required numbers
static void reachTheNums(int nums[], int k, int d1, int d2, int n)
{
    int i;
    int ans[] = new int[n];
    int gcd = __gcd(d1, d2);
 
    for (i = 0; i < n; i++) {
        int x = nums[i] - k;
 
        // If distance x is coverable
        if (x % gcd == 0)
            ans[i] = 1;
        else
            ans[i] = 0;
    }
 
    for (i = 0; i < n; i++)
        System.out.print(ans[i] + " ");
}
 
// Driver code
 
 
    public static void main (String[] args) {
        // Numbers to be checked for reachability
    int nums[] = { 9, 4 };
    int n =nums.length;
    // Starting number K
    int k = 8;
 
    // Sizes of jumps d1 and d2
    int d1 = 3, d2 = 2;
 
    reachTheNums(nums, k, d1, d2, n);
    }
}
 
// This code is contributed by inder_verma..

Python3

# Python3 implementation of the approach
import math as mt
 
# Function that returns the vector 
# containing the result for the reachability 
# of the required numbers
def reachTheNums(nums, k, d1, d2, n):
 
    ans = [0 for i in range(n)]
 
    gcd = mt.gcd(d1, d2)
 
    for i in range(n):
        x = nums[i] - k
 
        # If distance x is coverable
        if (x % gcd == 0):
            ans[i] = 1
        else:
            ans[i] = 0
 
    for i in range(n):
        print(ans[i], end = " ") 
 
# Driver code
 
# Numbers to be checked for
# reachability
nums = [9, 4]
n = len(nums)
 
# Starting number K
k = 8
 
# Sizes of jumps d1 and d2
d1, d2 = 3, 2
 
reachTheNums(nums, k, d1, d2, n)
 
# This code is contributed 
# by mohit kumar 29

C#

// C# implementation of the above approach 
 
using System ;
 
class GFG { 
     
    // Recursive function to return gcd of a and b 
    static int __gcd(int a, int b) 
    { 
        // Everything divides 0 
        if (a == 0) 
        return b; 
        if (b == 0) 
        return a; 
         
        // base case 
        if (a == b) 
            return a; 
         
        // a is greater 
        if (a > b) 
            return __gcd(a-b, b); 
             
        return __gcd(a, b-a); 
    } 
 
 
     
 
    // Function that returns the vector containing the 
    // result for the reachability of the required numbers 
    static void reachTheNums(int []nums, int k, int d1, int d2, int n) 
    { 
        int i; 
        int []ans = new int[n]; 
        int gcd = __gcd(d1, d2); 
     
        for (i = 0; i < n; i++) { 
            int x = nums[i] - k; 
     
            // If distance x is coverable 
            if (x % gcd == 0) 
                ans[i] = 1; 
            else
                ans[i] = 0; 
        } 
     
        for (i = 0; i < n; i++) 
            Console.Write(ans[i] + " "); 
    } 
 
    // Driver code 
    public static void Main () { 
        // Numbers to be checked for reachability 
    int []nums = { 9, 4 }; 
    int n =nums.Length; 
    // Starting number K 
    int k = 8; 
 
    // Sizes of jumps d1 and d2 
    int d1 = 3, d2 = 2; 
 
    reachTheNums(nums, k, d1, d2, n); 
    } 
    // This code is contributed by Ryuga 
} 

PHP

<?php
// PHP implementation of the approach
// gcd function
function GCD($a, $b)
{
    if ($b == 0) 
        return $a; 
    return GCD($b, $a % $b); 
}
 
// Function that returns the vector 
// containing the result for the
// reachability of the required numbers
function reachTheNums($nums, $k, $d1,
                             $d2, $n)
{
    $i = 0; $ans = array(0, 0);
    $gcd = GCD($d1, $d2);
 
    for ($i = 0; $i < $n; $i++) 
    {
        $x = $nums[$i] - $k;
 
        // if distance x is coverable
        if ($x % $gcd == 0)
            $ans[$i] = 1;
        else
            $ans[$i] = 0;
    }
 
    for ($i = 0; $i < $n; $i++)
    echo $ans[$i] . " ";
}
 
// Driver Code
 
// Numbers to be checked for reachability
$nums = array(9, 4);
$n = 2;
 
// Starting number $K
$k = 8;
 
// Sizes of jumps $d1 and $d2
$d1 = 3; $d2 = 2;
 
reachTheNums($nums, $k, $d1, $d2, $n);
     
// This code is contributed by Adesh Singh1 
?>

Javascript

<script>
// javascript implementation of the approach
 
    // Recursive function to return gcd of a and b
    function __gcd(a , b) 
    {
     
        // Everything divides 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;
 
        // base case
        if (a == b)
            return a;
 
        // a is greater
        if (a > b)
            return __gcd(a - b, b);
        return __gcd(a, b - a);
    }
 
    // Function that returns the vector containing the
    // result for the reachability of the required numbers
    function reachTheNums(nums , k , d1 , d2 , n) 
    {
        var i;
        var ans = Array(n).fill(0);
        var gcd = __gcd(d1, d2);
 
        for (let i = 0; i < n; i++)
        {
            var x = nums[i] - k;
 
            // If distance x is coverable
            if (x % gcd == 0)
                ans[i] = 1;
            else
                ans[i] = 0;
        }
 
        for (let i = 0; i < n; i++)
            document.write(ans[i] + " ");
    }
 
    // Driver code
     
    // Numbers to be checked for reachability
    var nums = [ 9, 4 ];
    var n = nums.length;
     
    // Starting number K
    var k = 8;
 
    // Sizes of jumps d1 and d2
    var d1 = 3, d2 = 2;
 
    reachTheNums(nums, k, d1, d2, n);
 
// This code is contributed by aashish1995. 
</script>

Output

1 1

Complexity Analysis:

Time Complexity: O(N), since there runs a loop from 0 to (n – 1).
Auxiliary Space: O(N), since N extra space has been taken.

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