# Find the pair (a, b) with minimum LCM such that their sum is equal to N

Given a number **N**, the task is to find two numbers a and b such that **a + b = N** and** LCM(a, b)** is minimum.

**Examples:**

Input:N = 15Output:a = 5, b = 10Explanation:The pair 5, 10 has a sum of 15 and their LCM is 10 which is the minimum possible.

Input:N = 4Output:a = 2, b = 2Explanation:The pair 2, 2 has a sum of 4 and their LCM is 2 which is the minimum possible.

**Approach:** The idea is to use the concept of GCD and LCM. Below are the steps:

- If
**N**is a Prime Number then the answer is**1**and**N – 1**because in any other cases either**a + b > N**or**LCM( a, b)**is**> N – 1**. This is because if**N**is prime then it implies that**N**is odd. So a and b, any one of them must be odd and other even. Therefore, LCM(a, b) must be greater than**N**( if not 1 and N â€“ 1) as 2 will always be a factor. - If
**N**is**not a prime number**then choose a, b such that their**GCD is maximum**, because of the formulaSo, in order to minimize LCM(a, b) we must**LCM(a, b) = a*b / GCD (a, b)**.maximize GCD(a, b). - If
**x**is a divisor of**N**, then by simple mathematics**N / x**and**N / x*( x – 1)**respectively. Now as**a = N / x**and**b = N / x * (x – 1)**, so their GCD comes out as**N / x**. To maximize this GCD, take the**smallest**possible x or smallest possible divisor of**N**.

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to check if number is` `// prime or not` `bool` `prime(` `int` `n)` `{` ` ` `// As 1 is neither prime` ` ` `// nor composite return false` ` ` `if` `(n == 1)` ` ` `return` `false` `;` ` ` `// Check if it is divided by any` ` ` `// number then it is not prime,` ` ` `// return false` ` ` `for` `(` `int` `i = 2; i * i <= n; i++) {` ` ` `if` `(n % i == 0)` ` ` `return` `false` `;` ` ` `}` ` ` `// Check if n is not divided` ` ` `// by any number then it is` ` ` `// prime and hence return true` ` ` `return` `true` `;` `}` `// Function to find the pair (a, b)` `// such that sum is N & LCM is minimum` `void` `minDivisor(` `int` `n)` `{` ` ` `// Check if the number is prime` ` ` `if` `(prime(n)) {` ` ` `cout << 1 << ` `" "` `<< n - 1;` ` ` `}` ` ` `// Now, if it is not prime then` ` ` `// find the least divisor` ` ` `else` `{` ` ` `for` `(` `int` `i = 2; i * i <= n; i++) {` ` ` `// Check if divides n then` ` ` `// it is a factor` ` ` `if` `(n % i == 0) {` ` ` `// Required output is` ` ` `// a = n/i & b = n/i*(n-1)` ` ` `cout << n / i << ` `" "` ` ` `<< n / i * (i - 1);` ` ` `break` `;` ` ` `}` ` ` `}` ` ` `}` `}` `// Driver Code` `int` `main()` `{` ` ` `int` `N = 4;` ` ` `// Function call` ` ` `minDivisor(N);` ` ` `return` `0;` `}` |

## Java

`// Java program for the above approach` `class` `GFG{` `// Function to check if number is` `// prime or not` `static` `boolean` `prime(` `int` `n)` `{` ` ` `// As 1 is neither prime` ` ` `// nor composite return false` ` ` `if` `(n == ` `1` `)` ` ` `return` `false` `;` ` ` `// Check if it is divided by any` ` ` `// number then it is not prime,` ` ` `// return false` ` ` `for` `(` `int` `i = ` `2` `; i * i <= n; i++)` ` ` `{` ` ` `if` `(n % i == ` `0` `)` ` ` `return` `false` `;` ` ` `}` ` ` `// Check if n is not divided` ` ` `// by any number then it is` ` ` `// prime and hence return true` ` ` `return` `true` `;` `}` `// Function to find the pair (a, b)` `// such that sum is N & LCM is minimum` `static` `void` `minDivisor(` `int` `n)` `{` ` ` `// Check if the number is prime` ` ` `if` `(prime(n))` ` ` `{` ` ` `System.out.print(` `1` `+ ` `" "` `+ (n - ` `1` `));` ` ` `}` ` ` `// Now, if it is not prime then` ` ` `// find the least divisor` ` ` `else` ` ` `{` ` ` `for` `(` `int` `i = ` `2` `; i * i <= n; i++)` ` ` `{` ` ` `// Check if divides n then` ` ` `// it is a factor` ` ` `if` `(n % i == ` `0` `)` ` ` `{` ` ` `// Required output is` ` ` `// a = n/i & b = n/i*(n-1)` ` ` `System.out.print(n / i + ` `" "` `+` ` ` `(n / i * (i - ` `1` `)));` ` ` `break` `;` ` ` `}` ` ` `}` ` ` `}` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `N = ` `4` `;` ` ` `// Function call` ` ` `minDivisor(N);` `}` `}` `// This code is contributed by Rajput-Ji` |

## Python3

`# Python3 program for the above approach` `# Function to check if number is` `# prime or not` `def` `prime(n):` ` ` ` ` `# As 1 is neither prime` ` ` `# nor composite return false` ` ` `if` `(n ` `=` `=` `1` `):` ` ` `return` `False` ` ` `# Check if it is divided by any` ` ` `# number then it is not prime,` ` ` `# return false` ` ` `for` `i ` `in` `range` `(` `2` `, n ` `+` `1` `):` ` ` `if` `i ` `*` `i > n:` ` ` `break` ` ` `if` `(n ` `%` `i ` `=` `=` `0` `):` ` ` `return` `False` ` ` `# Check if n is not divided` ` ` `# by any number then it is` ` ` `# prime and hence return true` ` ` `return` `True` `# Function to find the pair (a, b)` `# such that sum is N & LCM is minimum` `def` `minDivisor(n):` ` ` `# Check if the number is prime` ` ` `if` `(prime(n)):` ` ` `print` `(` `1` `, n ` `-` `1` `)` ` ` `# Now, if it is not prime then` ` ` `# find the least divisor` ` ` `else` `:` ` ` `for` `i ` `in` `range` `(` `2` `, n ` `+` `1` `):` ` ` `if` `i ` `*` `i > n:` ` ` `break` ` ` `# Check if divides n then` ` ` `# it is a factor` ` ` `if` `(n ` `%` `i ` `=` `=` `0` `):` ` ` `# Required output is` ` ` `# a = n/i & b = n/i*(n-1)` ` ` `print` `(n ` `/` `/` `i, n ` `/` `/` `i ` `*` `(i ` `-` `1` `))` ` ` `break` `# Driver Code` `N ` `=` `4` `# Function call` `minDivisor(N)` `# This code is contributed by mohit kumar 29` |

## C#

`// C# program for the above approach` `using` `System;` `class` `GFG{` `// Function to check if number is` `// prime or not` `static` `bool` `prime(` `int` `n)` `{` ` ` ` ` `// As 1 is neither prime` ` ` `// nor composite return false` ` ` `if` `(n == 1)` ` ` `return` `false` `;` ` ` `// Check if it is divided by any` ` ` `// number then it is not prime,` ` ` `// return false` ` ` `for` `(` `int` `i = 2; i * i <= n; i++)` ` ` `{` ` ` `if` `(n % i == 0)` ` ` `return` `false` `;` ` ` `}` ` ` `// Check if n is not divided` ` ` `// by any number then it is` ` ` `// prime and hence return true` ` ` `return` `true` `;` `}` `// Function to find the pair (a, b)` `// such that sum is N & LCM is minimum` `static` `void` `minDivisor(` `int` `n)` `{` ` ` `// Check if the number is prime` ` ` `if` `(prime(n))` ` ` `{` ` ` `Console.Write(1 + ` `" "` `+ (n - 1));` ` ` `}` ` ` `// Now, if it is not prime then` ` ` `// find the least divisor` ` ` `else` ` ` `{` ` ` `for` `(` `int` `i = 2; i * i <= n; i++)` ` ` `{` ` ` ` ` `// Check if divides n then` ` ` `// it is a factor` ` ` `if` `(n % i == 0)` ` ` `{` ` ` ` ` `// Required output is` ` ` `// a = n/i & b = n/i*(n-1)` ` ` `Console.Write(n / i + ` `" "` `+` ` ` `(n / i * (i - 1)));` ` ` `break` `;` ` ` `}` ` ` `}` ` ` `}` `}` `// Driver Code` `public` `static` `void` `Main(String[] args)` `{` ` ` `int` `N = 4;` ` ` `// Function call` ` ` `minDivisor(N);` `}` `}` `// This code is contributed by 29AjayKumar` |

## Javascript

`<script>` `// javascript program for the above approach ` `// Function to check if number is` ` ` `// prime or not` ` ` `function` `prime(n)` ` ` `{` ` ` ` ` `// As 1 is neither prime` ` ` `// nor composite return false` ` ` `if` `(n == 1)` ` ` `return` `false` `;` ` ` `// Check if it is divided by any` ` ` `// number then it is not prime,` ` ` `// return false` ` ` `for` `(i = 2; i * i <= n; i++)` ` ` `{` ` ` `if` `(n % i == 0)` ` ` `return` `false` `;` ` ` `}` ` ` `// Check if n is not divided` ` ` `// by any number then it is` ` ` `// prime and hence return true` ` ` `return` `true` `;` ` ` `}` ` ` `// Function to find the pair (a, b)` ` ` `// such that sum is N & LCM is minimum` ` ` `function` `minDivisor(n)` ` ` `{` ` ` `// Check if the number is prime` ` ` `if` `(prime(n))` ` ` `{` ` ` `document.write(1 + ` `" "` `+ (n - 1));` ` ` `}` ` ` `// Now, if it is not prime then` ` ` `// find the least divisor` ` ` `else` ` ` `{` ` ` `for` `(i = 2; i * i <= n; i++)` ` ` `{` ` ` `// Check if divides n then` ` ` `// it is a factor` ` ` `if` `(n % i == 0)` ` ` `{` ` ` `// Required output is` ` ` `// a = n/i & b = n/i*(n-1)` ` ` `document.write(n / i + ` `" "` `+ (n / i * (i - 1)));` ` ` `break` `;` ` ` `}` ` ` `}` ` ` `}` ` ` `}` ` ` `// Driver Code` ` ` `var` `N = 4;` ` ` `// Function call` ` ` `minDivisor(N);` `// This code is contributed by todaysgaurav` `</script>` |

**Output:**

2 2

**Time Complexity:**** **O(sqrt(N))**Auxiliary Space:** O(1)