# Check if it is possible to reach (X, Y) from (1, 1) by given steps

Given two integers **X** and **Y**, the task is to check if it is possible to reach **(X, Y)** from** (1, 1**) by the following possible moves:

- From a point
**(a, b)**such that**b > a**, move to the to point**(a, b – a)**. - From a point
**(a, b)**such that**a > b**, move to the to point**(a – b, b)**. - Move from any point
**(a, b)**to point**(2 * a, b)**or**(a, 2 * b)**.

If it is possible to reach **(X, Y), **print “**Yes**“. Otherwise, print **“No”**.

**Examples:**

Input:X = 4, Y = 7Output:YesExplanation:Point (4, 7) can be reached by the following steps: (1, 1) -> (1, 2) -> (1, 4) -> (1, 8) -> (1, 7) -> (2, 7) -> (4, 7)

Input:X = 3, Y = 6Output:No

**Naive Approach:** The simplest approach to solve the problem is to recursively check if **(1, 1) **can be reached from **(X, Y)** by making all possible moves from a point. If the point **(1, 1)** is reached at any point, print **“Yes”**. Otherwise, print **“No”**.

**Time Complexity:** O(N^{4}) where **N** is the number of steps to reach the point (1, 1).**Auxiliary Space: **O(1)

**Efficient Approach:** The idea is to find the greatest common divisor and observe the following facts:

- By moving the point from
**(a, b)**to the points**(a, b – a)**or**(a – b, b)**, there is no change in the GCD of the pair - By moving the point from
**(a, b)**to the points**(2 * a, b)**or**(a, 2 * b)**, GCD may get doubled or it may remain the same. - Therefore, from the above observations, point
**(1, 1)**can move to point**(X, Y)**if and only if gcd of**(X, Y)**is a power of**2**.

Follow the steps below to solve the above approach:

- Find the gcd of
**(X, Y)**and store it in a variable, say**val**. - Check if
**val**is a power of**2**by checking if**(val & amp;(val-1))**is equal to**0**where & is bitwise AND. - If it is a power of two print
**“Yes”**. Otherwise, print**“No”**.

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach ` `#include <iostream>` `using` `namespace` `std;` `// Function to find the gcd of ` `// two numbers` `int` `gcd(` `int` `a, ` `int` `b)` `{` ` ` ` ` `// Base case` ` ` `if` `(a < b) ` ` ` `{` ` ` `int` `t = a;` ` ` `a = b;` ` ` `b = t;` ` ` `}` ` ` `if` `(a % b == 0)` ` ` `return` `b;` ` ` ` ` `// Recurse` ` ` `return` `gcd(b, a % b);` `}` `// Function to print the answer` `void` `printAnswer(` `int` `x, ` `int` `y)` `{` ` ` ` ` `// GCD of X and Y` ` ` `int` `val = gcd(x, y);` ` ` `// If GCD is power of 2` ` ` `if` `((val & (val - 1)) == 0)` ` ` `cout << ` `"Yes"` `;` ` ` `else` ` ` `cout << ` `"No"` `;` `}` `// Driver code ` `int` `main()` `{` ` ` ` ` `// Given X and Y` ` ` `int` `x = 4;` ` ` `int` `y = 7;` ` ` `// Function call` ` ` `printAnswer(x, y);` ` ` `return` `0;` `}` `// This code is contributed by RohitOberoi` |

## Java

`// Java program for the above approach` `import` `java.io.*;` `class` `GFG {` ` ` `// Function to find the gcd of two numbers` ` ` `public` `static` `int` `gcd(` `int` `a, ` `int` `b)` ` ` `{` ` ` `// Base case` ` ` `if` `(a < b) {` ` ` `int` `t = a;` ` ` `a = b;` ` ` `b = t;` ` ` `}` ` ` `if` `(a % b == ` `0` `)` ` ` `return` `b;` ` ` `// Recurse` ` ` `return` `gcd(b, a % b);` ` ` `}` ` ` `// Function to print the answer` ` ` `static` `void` `printAnswer(` `int` `x, ` `int` `y)` ` ` `{` ` ` `// GCD of X and Y` ` ` `int` `val = gcd(x, y);` ` ` `// If GCD is power of 2` ` ` `if` `((val & (val - ` `1` `)) == ` `0` `)` ` ` `System.out.println(` `"Yes"` `);` ` ` `else` ` ` `System.out.println(` `"No"` `);` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `// Given X and Y` ` ` `int` `x = ` `4` `;` ` ` `int` `y = ` `7` `;` ` ` `// Function call` ` ` `printAnswer(x, y);` ` ` `}` `}` |

## Python3

`# Python3 program for the` `# above approach` `# Function to find the gcd ` `# of two numbers` `def` `gcd(a, b):` ` ` `# Base case` ` ` `if` `(a < b):` ` ` `t ` `=` `a` ` ` `a ` `=` `b` ` ` `b ` `=` `t` ` ` `if` `(a ` `%` `b ` `=` `=` `0` `):` ` ` `return` `b` ` ` `# Recurse` ` ` `return` `gcd(b, a ` `%` `b)` `# Function to print the ` `# answer` `def` `printAnswer(x, y):` ` ` `# GCD of X and Y` ` ` `val ` `=` `gcd(x, y)` ` ` `# If GCD is power of 2` ` ` `if` `((val & ` ` ` `(val ` `-` `1` `)) ` `=` `=` `0` `):` ` ` `print` `(` `"Yes"` `)` ` ` `else` `:` ` ` `print` `(` `"No"` `)` `# Driver code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `# Given X and Y` ` ` `x ` `=` `4` ` ` `y ` `=` `7` ` ` `# Function call` ` ` `printAnswer(x, y)` `# This code is contributed by Chitranayal` |

## C#

`// C# program for the above approach` `using` `System;` `class` `GFG{` `// Function to find the gcd of two numbers` `public` `static` `int` `gcd(` `int` `a, ` `int` `b)` `{` ` ` ` ` `// Base case` ` ` `if` `(a < b) ` ` ` `{` ` ` `int` `t = a;` ` ` `a = b;` ` ` `b = t;` ` ` `}` ` ` `if` `(a % b == 0)` ` ` `return` `b;` ` ` ` ` `// Recurse` ` ` `return` `gcd(b, a % b);` `}` `// Function to print the answer` `static` `void` `printAnswer(` `int` `x, ` `int` `y)` `{` ` ` ` ` `// GCD of X and Y` ` ` `int` `val = gcd(x, y);` ` ` `// If GCD is power of 2` ` ` `if` `((val & (val - 1)) == 0)` ` ` `Console.WriteLine(` `"Yes"` `);` ` ` `else` ` ` `Console.WriteLine(` `"No"` `);` `}` `// Driver code` `public` `static` `void` `Main()` `{` ` ` ` ` `// Given X and Y` ` ` `int` `x = 4;` ` ` `int` `y = 7;` ` ` `// Function call` ` ` `printAnswer(x, y);` `}` `}` `// This code is contributed by bgangwar59` |

## Javascript

`<script>` `// JavaScript program for the above approach` `// Function to find the gcd of` `// two numbers` `function` `gcd(a, b)` `{` ` ` ` ` `// Base case` ` ` `if` `(a < b)` ` ` `{` ` ` `let t = a;` ` ` `a = b;` ` ` `b = t;` ` ` `}` ` ` `if` `(a % b == 0)` ` ` `return` `b;` ` ` ` ` `// Recurse` ` ` `return` `gcd(b, a % b);` `}` `// Function to print the answer` `function` `printAnswer(x, y)` `{` ` ` ` ` `// GCD of X and Y` ` ` `let val = gcd(x, y);` ` ` `// If GCD is power of 2` ` ` `if` `((val & (val - 1)) == 0)` ` ` `document.write(` `"Yes"` `);` ` ` `else` ` ` `document.write(` `"No"` `);` `}` `// Driver code` ` ` ` ` `// Given X and Y` ` ` `let x = 4;` ` ` `let y = 7;` ` ` `// Function call` ` ` `printAnswer(x, y);` `// This code is contributed by Surbhi Tyagi.` `</script>` |

**Output:**

Yes

**Time Complexity:** O(Log(min(X, Y))) where (X, Y) is the given point. **Auxiliary Space:** O(1)