Question

Let n$\in \mathbb{N}$, and let $\overline{a}$ $\in \mathbb{Z}$_n be a reduced residue. Let r = odd($\overline{a}$). Prove that if r = st for positive integers s and t, then old($\overline{a}$^t) = s.

Answer 1

Solution given that het nEN, and lat a Ein be a reduced residue.. Let or= codla). we know that an be the set of residue classes 2012 Fi} and it foms a group onder addition modulo, defined by ā to atb. we know that it (Go) is a finite group, where o is the binary composition on G. and a EG such that ord(a) = n then [ord (an) = gold min where, an=a.a.a...a (in times) Now, 〠Zn and doord(a). also =st for some positive integence sandt. ast divides & so ged (tr) = t. Nowo, ord (at) = ord (2) god ſo (a), 2} - god (ast) - [ ord (at) = 5