Select Page

# Make two numbers equal in at most K steps dividing by their factor

Kashish Kumar
Published: June 17, 2022

Given integers X, Y, and K, the task is to make X and Y equal in not more than K operations by applying the following operations:

• Choose an integer A and make X = X/A, (where A is an integer that is greater than 1 and not equal to X).
• Choose an integer A and make Y = Y/A, (where A is an integer that is greater than 1 and not equal to Y).

Examples:

Input: X = 10, Y = 20, K = 4
Output: YES
Explanation: One optimal solution to make them equal is:
First choose A = 2 and do X = X/2 so now X is equal to 5.
Now choose A = 4 and do Y = Y/4 so now Y is equal to 5.
Since we have applied only two operations here which is less than K
to make X and Y equal and also greater than one.
Therefore the answer is YES.

Input: X = 2, Y = 27, K = 1
Output: NO
Explanation: There is no possible way to make X and Y equal
in less than or equal to 1 Operation.

Approach: To solve the problem follow the below idea:

Here are only two cases possible:

• When K is equal to one and
• When K is greater than 1

If K is equal to one then it is possible to make X and Y equal only when either X is divisible by Y or Y is divisible by X

If K is greater than 1 then X and Y can be equal and greater than 1 only when their GCD is greater than 1.

Follow the steps mentioned below to implement the idea:

• Check if K = 1:
• In that case, find out if any of them is a divisor of the other.
• Otherwise, find the GCD of the numbers.
• If the GCD is 1 then it is not possible.
• Otherwise, an answer always exists.

Below is the implementation for the above approach:

## C++

 ` ` `#include ``using` `namespace` `std;`` ` `bool` `isPossible(``int` `X, ``int` `Y, ``int` `K)``{``    ``    ``if` `(K == 1) {``        ``if` `(X % Y == 0 || Y % X == 0)``            ``return` `true``;``        ``return` `false``;``    ``}`` ` `    ``    ``else` `{``        ``if` `(__gcd(X, Y) > 1)``            ``return` `true``;``        ``return` `false``;``    ``}``    ``return` `false``;``}`` ` `int` `main()``{``    ``int` `X = 10;``    ``int` `Y = 20;``    ``int` `K = 4;`` ` `    ``    ``bool` `answer = isPossible(X, Y, K);``    ``if` `(answer)``        ``cout << ``"YES"``;``    ``else``        ``cout << ``"NO"``;``    ``return` `0;``}`

Time Complexity: O(1)
Auxiliary Space: O(1)

Source: www.geeksforgeeks.org