a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse
modulo n is well defined as a residue class in Zn, and depends only
on the residue class a in Zn.
b) Let Z × n ⊆ Zn be the set of invertible residue classes
modulo n. Prove that Z × n forms a group under multiplication. Is
this group a subgroup of Zn?
c) List the elements of Z × 9 . How many...
Let “ ·n” be multiplication modulo n, and consider the set Un =
{ [a] ∈ Zn | there is a [b] ∈ Zn with [a] ·n [b] = [1]}
(a) Show that (Un, ·n ) is a group.
(b) Write down the Cayley table for U5. Hint: |U5| = 4.
(c) Write down the Cayley table for U12. Hint: |U12| = 4.
a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 ). Your answer must
clearly specify the constants c and n0.
b) Let g(n) = 27n 2 + 18n and let f(n) = 0.5n 2 − 100. Find
positive constants n0, c1 and c2 such that c1f(n) ≤ g(n) ≤ c2f(n)
for all n ≥ n0. Be sure to explain how you arrived at the
constants.
Prove that for every n ∈ N:
a) (10^n + 3 * 4^(n+2)) ≡ 4 mod 19, [note that 4^3 ≡ 1 mod
9]
b) 24 | (2*7^(n) + 3*5^(n) - 5),
c) 14 | (3^(4n+2) + 5^(2n+1) [Note that 3^(4n+2) + 5^(2n+1) =
9^(2n)*9 + 5^(2n)*5 ≡ (-5)^(2n) * 9 + 5^(2n) *5 ≡ 0 mod 14]