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Stuck on a Subring

  1. Oct 4, 2011 #1
    Let S be a subring of the ring of integers ℤ. Show that if 13 ∈ S and 1000 ∈ S, then S equals ℤ.

    I'm thinking we should construct a bijection using both numbers. Any ideas? thanks
     
  2. jcsd
  3. Oct 4, 2011 #2
    I see that ∃ no takers. I know the problem looks weird but it is a problem I read in an old paper.

    The only thing I can think of is that we need to show that 1 ∈ S. To do this I can use 12 ∈ S because 12 = 1000 - 13*76, therefore 13 - 12 ∈ S and hence S has an identity and therefore we can extend to Z.
     
  4. Oct 4, 2011 #3

    lavinia

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    13 and 1000 are relatively prime so 1 is in the subring that they generate.
     
  5. Oct 4, 2011 #4
    Thank you Lavinia. You're becoming my favorite Science adviser. :)
     
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