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**1. Homework Statement**

Let Z

_{n}denote the set of integers {0,1,2,...n-1}. Let * be a binary operation on Z

_{n}such that a*b is the remainder of ab divided by n.

Show that (Z

_{n},*) is a semigroup for any n .

**2. Homework Equations**

**3. The Attempt at a Solution**

For showing the algebraic system to be a semi-group, we have to show that binary operation * is associative.

So if a,b,c ∈ Z

_{n},

We have to show, a * (b * c) =(a * b) * c

Now, a * (b * c) = a * rem(bc/n) (This is closed under Z

_{n})

= rem ( (a rem(bc/n) )/n)

Similarly, (a * b) * c = rem(ab/n) * c

=rem ( (rem(ab/n) c )/n)

Now , how to show here rem ( (a rem(bc/n) )/n) = rem ( (rem(ab/n) c )/n) ?

or a rem(bc/n) =rem(ab/n) c ?