Let us prove the converse first.
If t∣n . thus there exist k∈Z such that kt=n .
Now,
gn=gkt=(gt)k=ek=e Thus,
gn=e Suppose, t not divides n , hence by division algorithm there are p,r such that
n=pt+r,0<r<t Moreover we have given
gn=e⟹gn=gpt+r⟹(gt)pgr⟹epgr⟹egr=gr=e But we have given that t is given smallest , hence by definition t∣r⟹r≥t , hence we get a contradiction.
Therefore r must 0, hence
n=pt⟹t∣n We are done.
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