# Maximize profit when divisibility by two numbers have associated profits

Given five integers **N**, **A**, **B**, **X** and **Y**. The task is to find the maximum profit obtained from the numbers from the range **[1, N]**. If a positive number is divisible by **A** then the profit increases by **X** and if a positive number is divisible by **B** then the profit increases by **Y**.

**Note:** Profit from a positive number can be added at most once.

**Examples:**

Input:N = 3, A = 1, B = 2, X = 3, Y = 4

Output:10

1, 2 and 3 are divisible by A.

2 is the only number in the given range which is divisible by B.

2 is divisible by both A and B.

1 and 3 can be divided by A to get the profit of 2 * 3 = 6

2 can be divided by B to get the profit of 1 * 4 = 4

2 is divisible by both but in order to maximise the profit it is divided by B instead of A.

Input:N = 6, A = 6, B = 2, X = 8, Y = 2

Output:12

**Approach:** Easy to see that we can divide a number with both **A** and **B** only if the number is a multiple of **lcm(A, B)**. Obviously, that number should be divided with the number that gives more profit.

So the answer equals to **X * (N / A) + Y * (N / B) – min(X, Y) * (N / lcm(A, B))**.

Below is the implementation of the above approach:

## C++

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return the maximum profit ` `int` `maxProfit(` `int` `n, ` `int` `a, ` `int` `b, ` `int` `x, ` `int` `y) ` `{ ` ` ` `int` `res = x * (n / a); ` ` ` `res += y * (n / b); ` ` ` ` ` `// min(x, y) * n / lcm(a, b) ` ` ` `res -= min(x, y) * (n / ((a * b) / __gcd(a, b))); ` ` ` `return` `res; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 6, a = 6, b = 2, x = 8, y = 2; ` ` ` `cout << maxProfit(n, a, b, x, y); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java implementation of the approach ` `class` `GFG ` `{ ` ` ` `static` `int` `__gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(b == ` `0` `) ` ` ` `return` `a; ` ` ` `return` `__gcd(b, a % b); ` ` ` `} ` ` ` `// Function to return the maximum profit ` `static` `int` `maxProfit(` `int` `n, ` `int` `a, ` `int` `b, ` ` ` `int` `x, ` `int` `y) ` `{ ` ` ` `int` `res = x * (n / a); ` ` ` `res += y * (n / b); ` ` ` ` ` `// min(x, y) * n / lcm(a, b) ` ` ` `res -= Math.min(x, y) * (n / ((a * b) / __gcd(a, b))); ` ` ` `return` `res; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main (String[] args) ` `{ ` ` ` `int` `n = ` `6` `, a = ` `6` `, b = ` `2` `, x = ` `8` `, y = ` `2` `; ` ` ` `System.out.println(maxProfit(n, a, b, x, y)); ` `} ` `} ` ` ` `// This code is contributed by mits ` |

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## Python3

`# Python3 implementation of the approach ` `from` `math ` `import` `gcd ` ` ` `# Function to return the maximum profit ` `def` `maxProfit(n, a, b, x, y) : ` ` ` ` ` `res ` `=` `x ` `*` `(n ` `/` `/` `a); ` ` ` `res ` `+` `=` `y ` `*` `(n ` `/` `/` `b); ` ` ` ` ` `# min(x, y) * n / lcm(a, b) ` ` ` `res ` `-` `=` `min` `(x, y) ` `*` `(n ` `/` `/` `((a ` `*` `b) ` `/` `/` ` ` `gcd(a, b))); ` ` ` `return` `res; ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `n ` `=` `6` `;a ` `=` `6` `; b ` `=` `2` `; x ` `=` `8` `; y ` `=` `2` `; ` ` ` ` ` `print` `(maxProfit(n, a, b, x, y)); ` ` ` `# This code is contributed by Ryuga ` |

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## C#

`// C# implementation of the approach ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `static` `int` `__gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(b == 0) ` ` ` `return` `a; ` ` ` `return` `__gcd(b, a % b); ` ` ` `} ` ` ` `// Function to return the maximum profit ` `static` `int` `maxProfit(` `int` `n, ` `int` `a, ` `int` `b, ` `int` `x, ` `int` `y) ` `{ ` ` ` `int` `res = x * (n / a); ` ` ` `res += y * (n / b); ` ` ` ` ` `// min(x, y) * n / lcm(a, b) ` ` ` `res -= Math.Min(x, y) * (n / ((a * b) / __gcd(a, b))); ` ` ` `return` `res; ` `} ` ` ` `// Driver code ` `static` `void` `Main() ` `{ ` ` ` `int` `n = 6, a = 6, b = 2, x = 8, y = 2; ` ` ` `Console.WriteLine(maxProfit(n, a, b, x, y)); ` `} ` `} ` ` ` `// This code is contributed by mits ` |

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## PHP

**Output:**

12

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