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2 Keys Keyboard

Difficulty: Medium

Problem Description

There is only one character 'A' on the screen of a notepad. You can perform one of two operations on this notepad for each step:

  • Copy All: You can copy all the characters present on the screen (a partial copy is not allowed).
  • Paste: You can paste the characters which are copied last time.

Given an integer n, return the minimum number of operations to get the character 'A' exactly n times on the screen.

Key Insights

  • The problem can be approached by considering the factors of n.
  • Each time you want to reach n 'A's, you can think of how many times you can copy and paste based on the divisors of n.
  • The optimal way to reach a number is to use the least number of operations by using the copy operation strategically.

Space and Time Complexity

Time Complexity: O(sqrt(n))
Space Complexity: O(1)

Solution

To solve the problem, we can utilize a mathematical approach where we consider the factors of n. The algorithm involves iterating through all numbers from 1 to the square root of n. For each divisor found, we can compute the number of operations needed to create n 'A's. The total number of operations will be the sum of the found divisors.

Code Solutions

def minOperations(n):
    operations = 0
    # Iterate through possible divisors from 1 to sqrt(n)
    for i in range(1, int(n**0.5) + 1):
        if n % i == 0:  # i is a divisor
            operations += i  # Count the operations for the divisor
            if i != n // i:  # Check if there is a complementary divisor
                operations += n // i  # Count operations for the complementary divisor
    return operations - 1  # Subtract 1 because the initial 'A' is not an operation

# Example usage
print(minOperations(3))  # Output: 3
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