Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00 Edition Edition
ISBN: 9780395977224
Author: Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher: McDougal Littell
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![画像:Use the Euclidean algorithm to find two sets of integers (a, b, c) such that 55a65b+143c: Solution = 1. By the Euclidean algorithm, we have: 143 = 2.65 + 13 and 65 = 5.13, so 13 = 143 – 2.65. - Also, 55 = 4.13+3, 13 = 4.3 + 1 and 3 = 3.1, so 1 = 13 — 4.3 = 13 — 4(55 – 4.13) = 17.13 – 4.55. Combining these, we have: 1 = 17(143 – 2.65) - 4.55 = −4.55 - 34.65 + 17.143, so we can take a = − −4, b = −34, c = 17. By carrying out the division algorithm in other ways, we obtain different solutions, such as 19.55 23.65 +7.143, so a = = 9, b -23, c = 7. = = how ? come [Note that 13.55 + 11.65 - 10.143 0, so we can obtain new solutions by adding multiples of this equation, or similar equations.]](https://content.bartleby.com/qna-images/question/a6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4/b9c06ae5-fcee-41d4-9b51-dca71fbcfb7f/tktg7av_thumbnail.jpeg){kind=link}
Transcribed Image Text:Use the Euclidean algorithm to find two sets of integers (a, b, c) such
that
55a65b+143c:
Solution
= 1.
By the Euclidean algorithm, we have:
143 = 2.65 + 13 and 65 = 5.13, so 13 = 143 – 2.65.
-
Also, 55 = 4.13+3, 13 = 4.3 + 1 and 3 = 3.1,
so 1 = 13 — 4.3 = 13 — 4(55 – 4.13) = 17.13 – 4.55.
Combining these, we have:
1 = 17(143 – 2.65) - 4.55 = −4.55 - 34.65 + 17.143,
so we can take a = − −4, b = −34, c = 17. By carrying out the division
algorithm in other ways, we obtain different solutions, such as
19.55 23.65 +7.143, so a = = 9, b -23, c = 7.
=
=
how
?
come
[Note that 13.55 + 11.65 - 10.143 0, so we can obtain new solutions by
adding multiples of this equation, or similar equations.]
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