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Suppose n is an integer. Prove by direct method: If n is an odd integer then 8|(n² -1).

Suppose n is an integer. Prove by direct method: If n is an odd integer then 8|(n² -1).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 25E: 25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly...
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Prove the following theorems. Strictly follow instructions as to which method of proof will be used. Always write down what your assumptions are and what you need to show before your proof. You are allowed to directly use algebraic manipulations such as the use of special products, factoring, cross multiplication, transpose, substitution in your proof. Provide a justification for each line of your proof except those lines where you simply performed algebraic manipulations. You may use any of the definitions, axioms, and theorems as justification. State the theorem if the theorem has no name. You may also use the theorems on the photo for the proof.
Suppose n is an integer. Prove by direct method: If n is an odd integer then 8|(n2 -1).
Transcribed Image Text:Suppose n is an integer. Prove by direct method: If n is an odd integer then 8|(n2 -1).
Theorems (i), (iia), and (iib) can be used as justifications of a line of your proof. Let El=set of even integers and OI=set of odd integers. Theorem (i): If n is an integer, then neEI or nɛOI. Let n be any integer. Theorem (iia): n e El if and only if ngOI; or equivalently, Theorem (iib): n e OI if and only if ngEI.
Transcribed Image Text:Theorems (i), (iia), and (iib) can be used as justifications of a line of your proof. Let El=set of even integers and OI=set of odd integers. Theorem (i): If n is an integer, then neEI or nɛOI. Let n be any integer. Theorem (iia): n e El if and only if ngOI; or equivalently, Theorem (iib): n e OI if and only if ngEI.
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