Note: It is possible to prove Proposition 3.

Following are proofs for the first and fourth cases. Chunking — also known as the partial quotients method or the hangman method — is a less-efficient form of long division which may be easier to understand. Although this result doesn't seem too profound, it is nonetheless quite handy. If so, determine the remainder and prove that your answer is correct.

Use these two cases to prove the following proposition: Answer Add texts here. Recall that this is the idea used repeatedly when performing long division by hand.

Case 1. Now we're getting somewhere. Compare this to the proof of the same result in Proposition 3. We begin by squaring some integers that are not congruent to 0 modulo 5. Now, suppose that you have a pair of integers a and b, and would like to find the corresponding q and r.

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Section The Division Algorithm