Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an integer. Lemma 1.23:

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
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Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an integer.

Lemma 1.23:

 

Lemma 1.23: Let a, b e Z. If a and b are expressible in the form 4n + 1
where n is an integer, then ab is also expressible in that form.
Proof: Let a = 4n, + 1 and b
b =
4nz + 1 with п, пz€Z. Then
ab
(4n, + 1)(4n2 + 1) = 16n,n2 + 4n1 + 4n2 + 1
4(4n¡n2 + n1 + n2) + 1
=
4n + 1
where n =
4n,n2 + n, + n2ɛZ. I
Transcribed Image Text:Lemma 1.23: Let a, b e Z. If a and b are expressible in the form 4n + 1 where n is an integer, then ab is also expressible in that form. Proof: Let a = 4n, + 1 and b b = 4nz + 1 with п, пz€Z. Then ab (4n, + 1)(4n2 + 1) = 16n,n2 + 4n1 + 4n2 + 1 4(4n¡n2 + n1 + n2) + 1 = 4n + 1 where n = 4n,n2 + n, + n2ɛZ. I
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