a. Explanation Given information: The given expressions are The product of any two consecutive integer has the form 3 k or 3 k + 1 . Calculation: Quotient-remainder theorem: Given any integer n and positive integer d , there exist unique integers q and r such that n = d q + r and 0 ≤ r < d . Given d = 3 any integer n , begin by use the quotient-remainder theorem, n can be written in written in one of the three forms n = 3 q or n = 3 q + 1 or n = 3 q + 2 , for same integer q . Divide into following cases: Case (i): When n = 3 q Product of two consecutive integers can be written as n ( n + 1 = ( 3 q ) ( 3 q + 1 ) = 9 q 2 + 3 q = 3 k Where k = 3 q 2 + q Clearly as q is an integer, k = 3 q 2 + q is also an integer b. To determine To find: The in the form of mod notation.

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ISBN: 9781133417071

Author: EPP

Publisher: CENGAGE LEARNING - CONSIGNMENT

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Chapter 4.4, Problem 30ES

To determine

To find: The proof that product of any two consecutive integer has the form 3k or 3k+1 .

To determine

To find: The in the form of mod notation.

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Chapter 4.4, Problem 29ES

Chapter 4.4, Problem 31ES

Chapter 4 Solutions

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The proof that product of any two consecutive integer has the form 3 k or 3 k + 1 .

Solution Summary: The author explains the quotient-remainder theorem, wherein n can be written in one of the three forms.

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To find: The proof that product of any two consecutive integer has the form 3k or 3k+1 .

To find: The in the form of mod notation.

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Chapter 4 Solutions