Given an integer n, break it into the sum of k positive integers, where k >= 2, and maximize the product of those integers.Return the maximum product you can get.
Question
Given an integer n, break it into the sum of k positive integers, where k >= 2, and maximize the product of those integers.Return the maximum product you can get.
Solution
This problem can be solved using a dynamic programming approach. Here are the steps:
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Create an array dp of size n+1 and initialize all elements to 0. This array will store the maximum product for each integer from 0 to n.
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Set dp[1] = 1 because the maximum product of 1 when broken into the sum of at least two positive integers is 1.
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For each integer i from 2 to n, iterate through each integer j from 1 to i/2. For each j, calculate the product of j and the maximum of (i-j) and dp[i-j]. This represents the maximum product when i is broken into a sum of j and (i-j).
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Update dp[i] with the maximum product calculated in step 3.
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After filling up the dp array, dp[n] will be the maximum product when n is broken into the sum of at least two positive integers.
Here is a Python solution based on the above steps:
def maxProduct(n):
dp = [0]*(n+1)
dp[1] = 1
for i in range(2, n+1):
for j in range(1, i//2 + 1):
dp[i] = max(dp[i], j*max(i-j, dp[i-j]))
return dp[n]
This function takes an integer n as input and returns the maximum product when n is broken into the sum of at least two positive integers.
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