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# 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).

##### Solution Summary

The solution determines the nonnegative integers in the problem.

##### Solution Preview

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.

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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