MTH 617 assignment 1

Question 1 Let p be a prime with p ≥ 5. Let S := {`2,3,...,p−2`}. Show that for any x ∈ S, there is a y ∈ S with y 6= x such that xy ≡ 1(modp).

Deduce that

(p − 2)! ≡ 1(modp).

Question 2 Let n be any odd positive integer. Is it true that

5|2016ⁿ + 617²ⁿ?

(Hint: The year ”2016” and the course number ”617” appearing in this question could simply be distractions. )

Question 3 Let. Show that G is a group under ordinary multiplication of real numbers.

Question 4 Let n be an integer with n ≥ 2. For a,b in Z, define

a ×_n b := [ab]_n.

Show that ×_n is associative on Z.