Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Textbook Question
Chapter 10, Problem 16A
Round the following decimals to the indicated number of decimal places.
16.0.099 (3 places)
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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.]
-
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
p-1
2
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
23
32
how come?
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
The set T is the subset of these residues exceeding
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Chapter 10 Solutions
Mathematics For Machine Technology
Ch. 10 - Find the decimal value of the distance A in Figure...Ch. 10 - Find the decimal value of the distance B in Figure...Ch. 10 - Find the decimal value of the distance C in Figure...Ch. 10 - Use Figure 10-4 to answer Exercises 4 and 5. All...Ch. 10 - Use Figure 10-4 to answer Exercises 4 and 5. All...Ch. 10 - Multiply 413234 .Ch. 10 - Round the following decimals to the indicated...Ch. 10 - Round the following decimals to the indicated...Ch. 10 - Round the following decimals to the indicated...Ch. 10 - Round the following decimals to the indicated...
Ch. 10 - Prob. 11ACh. 10 - Round the following decimals to the indicated...Ch. 10 - Prob. 13ACh. 10 - Round the following decimals to the indicated...Ch. 10 - Prob. 15ACh. 10 - Round the following decimals to the indicated...Ch. 10 - Prob. 17ACh. 10 - Express the common fractions as decimal fractions....Ch. 10 - Prob. 19ACh. 10 - Express the common fractions as decimal fractions....Ch. 10 - Prob. 21ACh. 10 - Express the common fractions as decimal fractions....Ch. 10 - Prob. 23ACh. 10 - Express the common fractions as decimal fractions....Ch. 10 - Prob. 25ACh. 10 - Express the common fractions as decimal fractions....Ch. 10 - Prob. 27ACh. 10 - Express the common fractions as decimal fractions....Ch. 10 - Prob. 29ACh. 10 - Five pieces are cut from the length of round stock...Ch. 10 - Prob. 31ACh. 10 - Express the following decimal fractions as common...Ch. 10 - Prob. 33ACh. 10 - Express the following decimal fractions as common...Ch. 10 - Prob. 35ACh. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Express the following decimal fractions as common...Ch. 10 - Solve the following. 53. In Figure 108, what...Ch. 10 - In Figure 109, what common fractional part of...Ch. 10 - What common fractional part of distance A is each...
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