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Count numbers which can be constructed using two numbers
• Difficulty Level : Medium
• Last Updated : 31 Mar, 2021

Given three positive integers x, y and n, the task is to find count of all numbers from 1 to n that can be formed using x and y. A number can be formed using x and y if we can get it by adding any number of occurrences of x and/or y.
Examples :

```Input  : n = 10, x = 2, y = 3
Output : 9
We can form 9 out of 10 numbers using 2 and 3
2 = 2, 3 = 3, 4 = 2+2, 5 = 2+3, 6 = 3+3
7 = 2+2+3, 8 = 3+3+2, 9 = 3+3+3
and 10 = 3+3+2+2.

Input  : n = 10, x = 5, y = 7
Output : 3
We can form 3 out of 10 numbers using 5 and 7
The numbers are 5, 7 and 10

Input  : n = 15, x = 5, y = 7
Output : 6
We can form 6 out of 10 numbers using 5 and 7.
The numbers are 5, 7, 10, 12, 14 and 15.

Input  : n = 15, x = 2, y = 4
Output : 7```

A simple solution is to write a recursive code that starts with 0 and makes two recursive calls. One recursive call adds x and other adds y. This way we count total numbers. We need to make sure a number is counted multiple times.
An efficient solution solution is to use a boolean array arr[] of size n+1. An entry arr[i] = true is going to mean that i can be formed using x and y. We initialize arr[x] and arr[y] as true if x and y are smaller than or equal to n. We start traversing the array from smaller of two numbers and mark all numbers one by one that can be formed using x and y. Below is the implementation.

## C++

 `// C++ program to count all numbers that can``// be formed using two number numbers x an y``#include``using` `namespace` `std;` `// Returns count of numbers from 1 to n that can be formed``// using x and y.``int` `countNums(``int` `n, ``int` `x, ``int` `y)``{``    ``// Create an auxiliary array and initialize it``    ``// as false. An entry arr[i] = true is going to``    ``// mean that i can be formed using x and y``    ``vector<``bool``> arr(n+1, ``false``);` `    ``// x and y can be formed using x and y.``    ``if` `(x <= n)``        ``arr[x] = ``true``;``    ``if` `(y <= n)``        ``arr[y] = ``true``;` `    ``// Initialize result``    ``int` `result = 0;` `    ``// Traverse all numbers and increment``    ``// result if a number can be formed using``    ``// x and y.``    ``for` `(``int` `i=min(x, y); i<=n; i++)``    ``{``        ``// If i can be formed using x and y``        ``if` `(arr[i])``        ``{``            ``// Then i+x and i+y can also be formed``            ``// using x and y.              ``            ``if` `(i+x <= n)``                ``arr[i+x] = ``true``;``            ``if` `(i+y <= n)``                ``arr[i+y] = ``true``;` `            ``// Increment result``            ``result++;``        ``}``    ``}``    ``return` `result;``}` `// Driver code``int` `main()``{``    ``int` `n = 15, x = 5, y = 7;``    ``cout << countNums(n, x, y);``    ``return` `0;``}`

## Java

 `// Java program to count all numbers that can``// be formed using two number numbers x an y` `class` `gfg{``// Returns count of numbers from 1 to n that can be formed``// using x and y.``static` `int` `countNums(``int` `n, ``int` `x, ``int` `y)``{``    ``// Create an auxiliary array and initialize it``    ``// as false. An entry arr[i] = true is going to``    ``// mean that i can be formed using x and y``    ``boolean``[] arr=``new` `boolean``[n+``1``];` `    ``// x and y can be formed using x and y.``    ``if` `(x <= n)``        ``arr[x] = ``true``;``    ``if` `(y <= n)``        ``arr[y] = ``true``;` `    ``// Initialize result``    ``int` `result = ``0``;` `    ``// Traverse all numbers and increment``    ``// result if a number can be formed using``    ``// x and y.``    ``for` `(``int` `i=Math.min(x, y); i<=n; i++)``    ``{``        ``// If i can be formed using x and y``        ``if` `(arr[i])``        ``{``            ``// Then i+x and i+y can also be formed``            ``// using x and y.            ``            ``if` `(i+x <= n)``                ``arr[i+x] = ``true``;``            ``if` `(i+y <= n)``                ``arr[i+y] = ``true``;` `            ``// Increment result``            ``result++;``        ``}``    ``}``    ``return` `result;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``15``, x = ``5``, y = ``7``;``    ``System.out.println(countNums(n, x, y));``}``}``// This code is contributed by mits`

## Python3

 `# Python3 program to count all numbers``# that can be formed using two number``# numbers x an y` `# Returns count of numbers from 1``# to n that can be formed using x and y.``def` `countNums(n, x, y):` `    ``# Create an auxiliary array and``    ``# initialize it as false. An``    ``# entry arr[i] = True is going to``    ``# mean that i can be formed using``    ``# x and y``    ``arr ``=` `[``False` `for` `i ``in` `range``(n ``+` `2``)]` `    ``# x and y can be formed using x and y.``    ``if``(x <``=` `n):``        ``arr[x] ``=` `True``    ``if``(y <``=` `n):``        ``arr[y] ``=` `True` `    ``# Initialize result``    ``result ``=` `0` `    ``# Traverse all numbers and increment``    ``# result if a number can be formed``    ``# using x and y.``    ``for` `i ``in` `range``(``min``(x, y), n ``+` `1``):` `        ``# If i can be formed using x and y``        ``if``(arr[i]):` `            ``# Then i+x and i+y can also``            ``# be formed using x and y.``            ``if``(i ``+` `x <``=` `n):``                ``arr[i ``+` `x] ``=` `True``            ``if``(i ``+` `y <``=` `n):``                ``arr[i ``+` `y] ``=` `True` `            ``# Increment result``            ``result ``=` `result ``+` `1` `    ``return` `result` `# Driver code``n ``=` `15``x ``=` `5``y ``=` `7``print``(countNums(n, x, y))` `# This code is contributed by``# Sanjit_Prasad`

## C#

 `// C# program to count all numbers that can``// be formed using two number numbers x an y` `using` `System;` `public` `class` `GFG{``    ``// Returns count of numbers from 1 to n that can be formed``// using x and y.``static` `int` `countNums(``int` `n, ``int` `x, ``int` `y)``{``    ``// Create an auxiliary array and initialize it``    ``// as false. An entry arr[i] = true is going to``    ``// mean that i can be formed using x and y``    ``bool` `[]arr=``new` `bool``[n+1];` `    ``// x and y can be formed using x and y.``    ``if` `(x <= n)``        ``arr[x] = ``true``;``    ``if` `(y <= n)``        ``arr[y] = ``true``;` `    ``// Initialize result``    ``int` `result = 0;` `    ``// Traverse all numbers and increment``    ``// result if a number can be formed using``    ``// x and y.``    ``for` `(``int` `i=Math.Min(x, y); i<=n; i++)``    ``{``        ``// If i can be formed using x and y``        ``if` `(arr[i])``        ``{``            ``// Then i+x and i+y can also be formed``            ``// using x and y.            ``            ``if` `(i+x <= n)``                ``arr[i+x] = ``true``;``            ``if` `(i+y <= n)``                ``arr[i+y] = ``true``;` `            ``// Increment result``            ``result++;``        ``}``    ``}``    ``return` `result;``}` `// Driver code``    ``static` `public` `void` `Main (){``        ``int` `n = 15, x = 5, y = 7;``        ``Console.WriteLine(countNums(n, x, y));``    ``}``}`

## PHP

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## Javascript

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Output :

`6`

Time Complexity: O(n)
Auxiliary Space: O(n)
This article is contributed by Shivam Pradhan(anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
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