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2. Let o(a) and assume that a=bk Now, Let La) = <b> Then, we have to and that prove prime.
К. are relatively As La>= Lb> and o(a)=n implies o (6) Now, we Know that (6) ged d (n. k) O g cd (n, k) g cd (nk) g cd (n. k) = 1 which and K але relatively shows that prime.
and K are relatively Conversely, let prime.
Then ged (n,k) = 1 Given that o(a)=n and a=bk. Then, have to we prove that La>=l6> First of all, we will prove o(a) = o(b). To find o(b), we (bk) already know that o(b) ged (nak) o(6) ged (nak) o ca) o (6)=n [as ged (h, k)=1 Then, an=6=e. ΝΓοω, Let a & Ca> for some oLp an Now, (bk) KP b <b> So, La> s L67 Then In Now, EZ: as god (n , k) = 1 nay na tky=1
Now, for some osman Let b € <b> Then 6m. 1 = b m.
.
(natky) mx.n. mky b b my (bk) mx e amy amy a € La> so, 6m E La> Therefore, <b> t Las Sos we get that o(a)=0 (b) =n and and <b> S La> с <b> La) TV -La> = <b> Therefore, Hence, proved.
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