# Print all proper fractions with denominators less than equal to N

Given an integer N, the task is to print all proper fractions such that the denominator is less than or equal to N.

Proper Fractions: A fraction is said to be a proper fraction if the numerator is less than the denominator.

Examples:

Input: N = 3
Output: 1/2, 1/3, 2/3
Input: N = 4
Output: 1/2, 1/3, 1/4, 2/3, 3/4

Approach:
Traverse all numerators over [1, N-1] and, for each of them, traverse over all denominators in the range [numerator+1, N] and check if the numerator and denominator are coprime or not. If found to be coprime, then print the fraction.
Below is the implementation of the above approach:

## C++14

 `// C++ program to implement the ` `// above approach ` `#include   ` `using` `namespace` `std; ` ` `  `// Function to print all ` `// proper fractions ` `void` `printFractions(``int` `n) ` `{ ` `    ``for` `(``int` `i = 1; i < n; i++) { ` `        ``for` `(``int` `j = i + 1; j <= n; j++) { ` ` `  `            ``// If the numerator and the ` `            ``// denominator are coprime ` `            ``if` `(__gcd(i, j) == 1) { ` ` `  `                ``string a = to_string(i); ` `                ``string b = to_string(j); ` ` `  `                ``cout << a + ``"/"` `+ b << ``", "``; ` `            ``} ` `        ``} ` `    ``} ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `n = 3; ` `    ``printFractions(n); ` `    ``return` `0; ` `} `

## Java

 `// Java program to implement the ` `// above approach ` `class` `GFG{ ` ` `  `// Function to print all ` `// proper fractions ` `static` `void` `printFractions(``int` `n) ` `{ ` `    ``for``(``int` `i = ``1``; i < n; i++)  ` `    ``{ ` `        ``for``(``int` `j = i + ``1``; j <= n; j++)  ` `        ``{ ` `             `  `            ``// If the numerator and the ` `            ``// denominator are coprime ` `            ``if` `(__gcd(i, j) == ``1``) ` `            ``{ ` `                ``String a = String.valueOf(i); ` `                ``String b = String.valueOf(j); ` ` `  `                ``System.out.print(a + ``"/"` `+  ` `                                 ``b + ``", "``); ` `            ``} ` `        ``} ` `    ``} ` `} ` ` `  `static` `int` `__gcd(``int` `a, ``int` `b)  ` `{  ` `    ``return` `b == ``0` `? a : __gcd(b, a % b);      ` `}  ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `n = ``3``; ` `     `  `    ``printFractions(n); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

## Python3

 `# Python3 program for the ` `# above approach ` ` `  `# Function to print ` `# all proper functions ` `def` `printfractions(n): ` `   `  `  ``for` `i ``in` `range``(``1``, n): ` `    ``for` `j ``in` `range``(i ``+` `1``, n ``+` `1``): ` `       `  `      ``# If the numerator and  ` `      ``# denominator are coprime ` `      ``if` `__gcd(i, j) ``=``=` `1``: ` `        ``a ``=` `str``(i) ` `        ``b ``=` `str``(j) ` `        ``print``(a ``+` `'/'` `+` `b, end ``=` `", "``) ` `         `  `def` `__gcd(a, b): ` `   `  `  ``if` `b ``=``=` `0``: ` `    ``return` `a ` `  ``else``: ` `    ``return` `__gcd(b, a ``%` `b) ` ` `  `# Diver code ` `if` `__name__``=``=``'__main__'``: ` `   `  `  ``n ``=` `3` `  ``printfractions(n) ` ` `  `# This code is contributed by virusbuddah_`

## C#

 `// C# program to implement the ` `// above approach ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `// Function to print all ` `// proper fractions ` `static` `void` `printFractions(``int` `n) ` `{ ` `    ``for``(``int` `i = 1; i < n; i++)  ` `    ``{ ` `        ``for``(``int` `j = i + 1; j <= n; j++)  ` `        ``{ ` `             `  `            ``// If the numerator and the ` `            ``// denominator are coprime ` `            ``if` `(__gcd(i, j) == 1) ` `            ``{ ` `                ``string` `a = i.ToString(); ` `                ``string` `b = j.ToString(); ` ` `  `                ``Console.Write(a + ``"/"` `+  ` `                              ``b + ``", "``); ` `            ``} ` `        ``} ` `    ``} ` `} ` ` `  `static` `int` `__gcd(``int` `a, ``int` `b)  ` `{  ` `    ``return` `b == 0 ? a : __gcd(b, a % b);      ` `}  ` ` `  `// Driver code ` `public` `static` `void` `Main(``string``[] args) ` `{ ` `    ``int` `n = 3; ` `     `  `    ``printFractions(n); ` `} ` `} ` ` `  `// This code is contributed by rutvik_56`

Output:

```1/2, 1/3, 2/3,
```

Time Complexity: O(N2
Auxiliary Space: O(1)

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