stuck on a proof..any help would be great!
show, for any integer k>=0:
en(a^k)= en(a) / gcd( en(a), k)
where en(a) is the order of a modulo n, also (a,n) = 1
so far I have: let 'b' be the order of a^k mod n
let 'h' be the order of a mod n
so (a^b*k) congr. a^h congr. 1 mod n
so h | bk
also, a^(h*k) congr. a^(b*k) congr. 1 mod n
rearranging the exponents and setting equal modulo b gives: b|h
so I have h | bk and b|h...is there somewhere to go from here?
show, for any integer k>=0:
en(a^k)= en(a) / gcd( en(a), k)
where en(a) is the order of a modulo n, also (a,n) = 1
so far I have: let 'b' be the order of a^k mod n
let 'h' be the order of a mod n
so (a^b*k) congr. a^h congr. 1 mod n
so h | bk
also, a^(h*k) congr. a^(b*k) congr. 1 mod n
rearranging the exponents and setting equal modulo b gives: b|h
so I have h | bk and b|h...is there somewhere to go from here?