
2. Let a,b,c E Z. Prove the following. If aſb then g.c.d(b, c) = 1 implies...
(1) Prove or disprove the following statements. (a) Let a, b and c be integers. If aſc and b|c, then (a + b)|c (b) Let a, b and c be integers. If aſb, then (ac)(bc)
Let a, b, c e Z with a + 0. If a|c, then there exists an integer b such that aſb and b|c.
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
(4 pts) Choose the true statement. Let a,b,c e Z with a #0. If a|(b + c) and a|bc, then a|(+ c´). Let a,b,c e Z with a +0. If a|(62 +c), then a|(b+c). Let a, b, ce Z with a + 0. If a|(b+c), then aſb or a|c. Let a,b,c e Z with a +0. If a|(62 +c) and a|(b+c?), then a|(+ c´). the four other possible answers are false
(3) If z = a + ib E C and |2| := Va² + b², prove that |zw| = |z||w]. Proof. Proof here. goes (4) Let y : C× → R* be defined by 9(z) = |z|. Use Problem (3) to prove that y is a homomorphism. Proof. Proof goes here.
3. Let a, b, c E Z such that ca and (a,b) = 1. Show that (c, b) = 1. 4. Suppose a, b, c, d, e E Z such that e (a - b) and e| (c,d). Show that e (ad — bc). 5. Fix a, b E Z. Consider the statements P: (a, b) = 1, and Q: there exists x, y E Z so that ax + by = 1. Bézout’s lemma states that: if P, then...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli
5. Let Zli_ {a + bi l a,b...
3. (6 points) Disprove the following: For all a, b, c E Z, if aſbc, then aſb or b|c.
0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT
0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT