## necessary and sufficient condition

Could someone please tell me what condition on n is necessary and sufficient such that Z/nZ contains a nilpotent element?

I can't seem to find this in any of my texts.

By the way, I accidently posted this same question in the homework help section first. Sorry to anyone who's wondering why I'm cluttering up multiple boards with my quandries.
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 Recognitions: Homework Help Science Advisor suppose that n is the product of distinct primes p_i, with multiplicity n_i (i ranges form 1 to r), then is p_1p_2...p_r niplotent? can you see how to use that idea to find a necessary condition?
 okay, so let's see.. you're asking me whether n itself is a nilpotent element of Z/nZ. yes, I think n = (p_1)(p_2)...(p_r) is nilpotent by definition because: n^k is congruent to 0 (mod n), for k > 0 ...right? I'm not sure how your hint regarding the prime factorization can be used though. Can you elaborate? I did manage to find something while I was reading that I thought might help me: if x^k is congruent to 0 (mod n) ---> then every prime dividing n divides x also. *this works as far as I can tell (just by trying a few examples) though I'm not sure why. but it seems to be along the same lines as your hint (because "every prime dividing n" is the prime factorization of n ..which you told me to consider).* so I guess all the nilpotent elements of Z/nZ would have to be divisible by every single prime which divides n. I still don't see how any of this leads me to a necessary condition for n though. please help!

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