421 HW 1 Individual
Prime factorization should not be used for proofs in this homework.
Problem (1.1.1).
Find the quotient q and remainder r when a is divided by b, without using technology. Check your answers.
(a) a = 17; b = 4 (b) a = 0; b = 19 (c) a = −17; b = 4
Solution：
Problem (1.1.3). 数学抽象代数代写
Use a calculator to find the quotient q and remainder r when a is divided by b.
(a) a = 517; b = 83 (b) a = −612; b = 74 (c) a = 7, 965, 532; b = 127
Solution：
Problem (1.1.7).
Prove that the square of any integer a is either of the form 3k or of the form 3k + 1 for some integer k. [Hint: By the Division Algorithm, a must be of the form 3q or 3q + 1 or 3q + 2.]
Solution：
Problem (1.2.4). 数学抽象代数代写
(b) If a | b and a | c, prove that a | (br + ct) for any r, t ∈ Z.
Solution：
Problem (1.2.5).
If a and b are nonzero integers such that a | b and b | a, prove that a = ±b.
Solution：
Problem (1.2.15). 数学抽象代数代写
The Euclidean Algorithm is an efficient way to find (a, b) for any positive integers a and b. It only requires you to apply the Division Algorithm several times until you reach the gcd. (See this exercise in the book for the example of (524, 148) and what parts (f)-(i) are about.)
Use the Euclidean Algorithm to find
(c) (1003, 456)
(j) Use the method described in parts (f)-(i) to express the gcd in part (c) as a linear combination of 1003 and 456.
Solution：
Problem (1.2.25(a)). 数学抽象代数代写
If a, b, u, v ∈ Z are such that au + bv = 1, prove that (a, b) = 1.
Solution：
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