# 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(mod*p*).

Deduce that

(*p *− 2)! ≡ 1(mod*p*)*.*

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

5|2016* ^{n }*+ 617

^{2n}?

(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**.