Say you want to show that then or . Well (3) is principal. So it is maximal. Thus the result follows? Are there any other ways of showing this?
Supose (3) < I ==> there's a non-multiple of 3 in I, say x ==> since 3 is prime, (x,3) = 1 ==> there exist integers m, n s.t. 3m + xn = 1.
But 3m in (3) < I and xn in I ==> 1 = 3m + xn in I ==> I = Z and we're done.