- #1
Raghav Gupta
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- 76
Homework Statement
Let Zn denote the set of integers {0,1,2,...n-1}. Let * be a binary operation on Zn such that a*b is the remainder of ab divided by n.
Show that (Zn,*) is a semigroup for any n .
Homework Equations
The Attempt at a Solution
[/B]For showing the algebraic system to be a semi-group, we have to show that binary operation * is associative.
So if a,b,c ∈ Zn,
We have to show, a * (b * c) =(a * b) * c
Now, a * (b * c) = a * rem(bc/n) (This is closed under Zn )
= 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 ?