MATH21112 Rings and Fields
Example Sheet 1 - Properties of Z and Zn
1. Find the greatest common divisor of 2827 and 374 and write the gcd as a linear combination of 2827 and 374.
2. Let n ∈ Z with n ≥ 2. Prove that 三 mod n has the following properties:
(i) 8a ∈ Z, a 三 a mod n.
(ii) 8a, b ∈ Z, if a 三 b mod n, then b 三 a mod n.
(iii) 8a,b, c ∈ Z, if a 三 b mod n and b 三 c mod n, then a 三 c mod n.
(We say that 三 mod n is an equivalence relation on the set of integers.)
3. Let a,b, n ∈ Z with n ≥ 2. Prove that a 三 b mod n if and only if [a]n = [b]n .
4. Suppose that a, n ∈ Z, with n ≥ 2, are coprime (ie. gcd(a, n) = 1). Show that there exists [b]n ∈ Zn such that [ab]n = [1]n.