# Determine the Nonnegative Integers

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Please see the attachment for problem related to nonnegative integer and my solution (needs to be edited and confirmed).

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##### Solution Summary

The solution determines the nonnegative integers in the problem.

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Please see attached file

The answer is correct, and clear, so I only made some changes (in bold) which I think may make more sense.

Estimated Time/Compensation: 30 min/$8

Date Needed/Deadline: Sunday, November 7, 2004. 1:00 P.M. Pacific

There is a solution that is already provided.

I ask that you check the solution correctness, and clarity.

If there is a proof, please do your best to explain the proof clearer, with words.

If you find an error , mistake, or something could be "more elegant" than please rework the entire solution.

PLEASE FOLLOW THE INSTRUCTIONS:

1. No programming

2. Show all steps

3. Explain your solution process in words

4. Proofs need to be explained primarily in words. If there is a lot of algebra, explain the process in words as well.

5. ANSWER ALL PARTS OF THE QUESTION

For every nonnegative integer n, there are unique nonnegative integers i and j such that

and

To see this, consider division of n by 43, and let i and j denote the quotient and the remainder, respectively:

j is an element of Z43, and n is said to be congruent modulo 43 to j, which we could express by saying that n is EQUIVALENT to j (modulo 43). (In YOUR notation, we would say that "n = j in Z43.")

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