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.
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