If a + b + c + d = 13, find the minimum possible value of a3 + b3 + c3 + d3, given that a, b, c and d are all positive integers

The problem is asking for the minimum possible value of a³ + b³ + c³ + d³, given that a, b, c, and d are all positive integers and their sum is 13.

Here are the steps to solve the problem:

Step 1: Since a, b, c, and d are all positive integers, the smallest value they can have is 1.

Step 2: To minimize a³ + b³ + c³ + d³, we should make as many of the variables as possible equal to 1, because 1³ = 1, which is the smallest cube of a positive integer.

Step 3: The smallest sum of four positive integers that equals 13 is 1 + 1 + 1 + 10 = 13. This corresponds to a = 1, b = 1, c = 1, and d = 10.

Step 4: Substituting these values into the expression a³ + b³ + c³ + d³ gives 1³ + 1³ + 1³ + 10³ = 1 + 1 + 1 + 1000 = 1003.

So, the minimum possible value of a³ + b³ + c³ + d³, given that a, b, c, and d are all positive integers and their sum is 13, is 1003.