We need to split a number n such that sum of maximum divisors of all the parts is minimum.
Input: n = 27 Output: Minimum sum of maximum divisors of parts = 3 Explanation : We can split 27 as follows: 27 = 13 + 11 + 3, Maximum divisor of 13 = 1, 11 = 1, 3 = 1. Answer = 3 (1 + 1 + 1). Input : n = 4 Output: Minimum sum of maximum divisors of parts = 2 Explanation : We can write 4 = 2 + 2 Maximum divisor of 2 = 1, So answer is 1 + 1 = 2.
We need to minimize maximum divisors. It is obvious that if N is prime, maximum divisor = 1. If the number is not a prime, then the number should be atleast 2.
According to Goldbach’s Conjecture, every even integer can be expressed as sum of two prime numbers. For our problem there will be two cases:
1) When the number is even, it can be expressed as the sum of two prime numbers and our answer will be 2 because maximum divisor of a prime number is 1.
2) When the number is odd, it can also be written as sum of prime numbers, n = 2 + (n-2); if (n-2) is a prime number(answer = 2), otherwise. Refer odd number as sum of primes for details.
n = 3 + (n-3); (n-3) is an even number and it is sum of two primes(answer = 3).
Below is the implementation of this approach.
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Improved By : jit_t