Question

Is there any mathematical exact prove online for pset2 problem 2?

Answer 1

I am taking a wild guess here, but I think that it's due to the property of divisibility of other solutions, i.e., pages 24-25 from http://www-groups.dcs.st-and.ac.uk/~martyn/teaching/1003/1003linearDiophantine.pdf , I will try to find a mathematical proof, but I think this gives an intuition of why it works, and ideas to work on the proof.

Answer 2

I think that you will be able to understand why this works by considering that there are 6 (equal to the smallest order size) fulfillable orders in a row. Maybe you can see how for any larger integer it could by written as one of these 6 values plus some even multiple of 6.
More formally, you would like to show that any integer k > x+5 can be written as x + offset + 6 * j, where 0<= offset <=5. So, to whatever solution you had for the number between x and x+5 just add j orders of size 6.

Note, that in the general case, the smallest order need not contain 6 pieces, but that is the number of consecutive solutions you need to be able to solve for all larger values.

I hope that helps.