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Homework Help: Abstract Algebra proof

  1. Jul 26, 2010 #1
    Let a, b be integers a,b>0 show that if a^3 | b^2 then a|b
    (Consider the prime factorization of a and b)



    I've tried setting up generic prime factorization of a and b but then dont get any where, i'm not very strong at this subject.

    Any kind of hints / where to start would help alot thanks!
     
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  3. Jul 26, 2010 #2

    Dick

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    You already have a perfectly good clue. How did you try to use prime factorization? Show the number of times any prime p divides a is less than or equal to the number of times p divides b.
     
    Last edited: Jul 26, 2010
  4. Jul 26, 2010 #3
    two things, how should i show a prime p, divides a ex.
    (p=2^e1 *3^e2 *5^e3....) = a*q?, q is an integer

    And why do i need to show that it is less than or equal to the number of times p divides b?
     
  5. Jul 26, 2010 #4

    Dick

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    Let's do the second one first. The only way a can divide b is if the number of times every prime p divides a is less or equal to the number of times p divides b. Think about the prime factorization of b/a. Don't you agree?
     
  6. Jul 26, 2010 #5

    yes i agree with what you are saying here.
     
  7. Jul 26, 2010 #6

    Dick

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    Well, ok. So then if the largest power of p in a is p^ka and the largest power of p in b is p^kb, what must be true if a^3 divides b^2?
     
  8. Jul 26, 2010 #7
    do you mean a^3 divides b^2? if so then, b would be larger than a?
     
  9. Jul 26, 2010 #8

    Dick

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    No, I'm asking you to compare the number of times p divides a^3 versus the number of times p divides b^2. The first must be less than or equal to the second, right? What does that tell you about ka and kb?
     
  10. Jul 26, 2010 #9
    kb must be bigger than ka?
     
  11. Jul 26, 2010 #10

    Dick

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    Yes. Why does a^3 divides b^2 tell you that? Please help me here. I can't just tell you what to write down. You have to understand it.
     
  12. Jul 26, 2010 #11
    is it that there is some factor times b that makes a=b?
     
  13. Jul 26, 2010 #12
    The best way to start is to just try some simple examples. Make up some small examples which you know will work / not work.

    For example, see what happens for a=15 and b=225 (use prime factorization, as hinted).

    Now try something like a=15 and b=75. Why doesn't this example work?

    See if you can then come up with the general idea.
     
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