Math 328K (Daniel Allcock) Homework 4, due Friday Feb. 19, 2016

Math 328K (Daniel Allcock)
Homework 4, due Friday Feb. 19, 2016
Problem 1. We have seen that Zm is a lot like Z, but can be strange in some ways
(like “even” not being meaningful when m is odd). This problem points out another
difference. I assume you know the fact that a polynomial of degree d has at most
d roots in the real numbers R. Well, the obvious analogue of this can fail in Zm :
(a) the polynomial equation x2 = 1 has more than two solutions in Z8 : find them
all, count them, and observe that there more than the degree of the polynomial.
(b) the same as (a) with Z8 replaced by Z15 .
(Curiously and deeply, it turns out that the fact above for polynomials does
hold in Zm if m is a prime number. We will see more of this later in the course.)
Problem 2. If a, b, c ∈ Z, then a|b if and only if ac|bc.
Problem 3. Suppose a, b ∈ Z and a|b. Prove that an |bn for every positive integer n.
Problem 4. Compute the following gcd’s. (More practice: no. 1 in section 3.3.)
(a) gcd(273, 91)
(b) gcd(10001, 1001)
Problem 5. (a) If k is an integer then gcd(6k + 7, 3k + 4) = 1.
(b) gcd(n + 1, n2 − n + 1) can have take two possible values, when n varies over
the integers. Find these values, and which n’s produce them.
Problem 6. For each of the following pairs of numbers, find the gcd, and find Bezout
numbers for them.
(a) (51, 87)
(b) (105, 300)
(c) (981, 1234)
(d) (34709, 100313)
Problem 7. For each of the following linear Diophantine equations, either find all
integer solutions or show that there are none.
(a) 3x + 4y = 7
(b) 12x + 18y = 50
(c) 30x + 47y = −11
(d) 25x + 95y = 970
(e) 102x + 1001y = 1