Given a range from L to R and every Xth tile is painted black and every Yth tile is painted white in that range from L to R. If a tile is painted both white and black, then it is considered to be painted grey. The task is to find the number of tiles that are colored grey in range L to R (both inclusive).
Examples:
Input: X = 2, Y = 3, L = 6, R = 18 Output: 3 The grey coloured tiles are numbered 6, 12, 18 Input: X = 1, Y = 4, L = 5, R = 10 Output: 1 The only grey coloured tile is 8.
Approach: Since every multiple of X is black and every multiple of Y is white. Any tile which is a multiple of both X and Y would be grey. The terms that are divisible by both X and Y are the terms that are divisible by the lcm of X and Y.
Lcm can be found out using the following formula:
lcm = (x*y) / gcd(x, y)
GCD can be computed in logn time using Euclid’s algorithm. The number of multiples of lcm in range L to R can be found by using a common trick of:
count(L, R) = count(R) - count(L-1)
Number of terms divisible by K less than N is:
floor(N/K)
Below is the implementation to find the number of grey tiles:
// C++ implementation to find the number of // grey tiles #include <bits/stdc++.h> using namespace std;
// Function to count the number of grey tiles int findTileCount( int x, int y, int l, int r)
{ int lcm = (x * y) / __gcd(x, y);
// Number multiple of lcm less than L
int countl = (l - 1) / lcm;
// Number of multiples of lcm less than R+1
int countr = r / lcm;
return countr - countl;
} // Driver code int main()
{ int x = 2, y = 3, l = 6, r = 18;
cout << findTileCount(x, y, l, r);
return 0;
} |
// Java implementation to find the // number of grey tiles import java.io.*;
class GFG {
// Function to count the number
// of grey tiles static int findTileCount( int x, int y,
int l, int r)
{ int lcm = (x * y) / __gcd(x, y);
// Number multiple of lcm less than L
int countl = (l - 1 ) / lcm;
// Number of multiples of
// lcm less than R+1
int countr = r / lcm;
return countr - countl;
} 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);
} // Driver code public static void main (String[] args) {
int x = 2 , y = 3 , l = 6 , r = 18 ;
System.out.println(findTileCount(x, y, l, r));
} } // This code is contributed ajit |
# Python3 implementation to find the number of # grey tiles # from math lib import gcd method from math import gcd
# Function to count the number of grey tiles def findTileCount(x, y, l, r) :
lcm = (x * y) / / gcd(x, y)
# Number multiple of lcm less than L
count1 = (l - 1 ) / / lcm
# Number of multiples of lcm less than R+1
countr = r / / lcm
return countr - count1
# Driver code if __name__ = = "__main__" :
x, y, l, r = 2 , 3 , 6 , 18
print (findTileCount(x, y, l, r))
# This code is contributed by # ANKITRAI1 |
// C# implementation to find the // number of grey tiles using System;
class GFG
{ // Function to count the number // of grey tiles static int findTileCount( int x, int y,
int l, int r)
{ int lcm = (x * y) / __gcd(x, y);
// Number multiple of lcm less than L
int countl = (l - 1) / lcm;
// Number of multiples of
// lcm less than R+1
int countr = r / lcm;
return countr - countl;
} 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);
} // Driver code public static void Main()
{ int x = 2, y = 3, l = 6, r = 18;
Console.Write(findTileCount(x, y, l, r));
} } // This code is contributed // by Kirti_Mangal |
<?php // PHP implementation to find the // number of grey tiles // Function to count the number // of grey tiles function findTileCount( $x , $y , $l , $r )
{ $lcm = (int)(( $x * $y ) / __gcd( $x , $y ));
// Number multiple of lcm less than L
$countl = (int)(( $l - 1) / $lcm );
// Number of multiples of
// lcm less than R+1
$countr = (int)( $r / $lcm );
return $countr - $countl ;
} 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 );
} // Driver code $x = 2; $y = 3; $l = 6; $r = 18;
echo findTileCount( $x , $y , $l , $r );
// This code is contributed // by Akanksha Rai(Abby_akku) ?> |
<script> // JavaScript implementation to find the // number of grey tiles // Function to count the number // of grey tiles function findTileCount(x,y,l,r)
{ lcm = parseInt((x * y) / __gcd(x, y));
// Number multiple of lcm less than L
countl = parseInt((l - 1) / lcm);
// Number of multiples of
// lcm less than R+1
countr = parseInt(r / lcm);
return countr - countl;
} 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);
} // Driver code let x = 2; let y = 3; let l = 6; let r = 18; document.write(findTileCount(x, y, l, r)); // This code is contributed by bobby </script> |
3
Time Complexity: O(log(min(x, y))), where x and y are two parameters of gcd.