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Math Help - Number Theory

  1. #1
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    Number Theory

    Let n and a be positive integers and let p be a prime number. The p^a is said to exactly divide n. If p^a exactly divides n and p^a+1 does not exactly divide n. Assume p^a exactly divides m and p^b exactly divides n.

    What power of p exactly divides m+n. Prove.
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  2. #2
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    Lemma: If p^a exactly divides n, then no greater power of p divides n, and conversely, if p^a | n and no greater power of p divides n, then p^a exactly divides n.

    The answer to your question is obviously a + b, but why? Can you use the lemma to show how?
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  3. #3
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    yes you can use the lemma.
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  4. #4
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    It is clear that if p^a | m and p^b | n, then p^{(a+b)} | mn. Suppose p^{(a+b+k)} | mn for some positive integer k and p^a, p^b both exactly divide m and n, respectively. Then since a is the greatest power of p that divides m, it must be true that p^{(b+k)} | n, which is impossible by the lemma. Therefore, p^{(a+b)} divides mn exactly.

    Edit: never mind, I had the question wrong.
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  5. #5
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    I don't think you have enough information.

    Consider the following series of integers m, n:

    m = p(p - 1), n = p, m + n = p^2

    m = p(p^2 - 1), n = p, m + n = p^3

    m = p(p^3 - 1), n = p, m + n = p^4

    etc.

    In each case it is clear that p exactly divides both m and n, but the power of p that exactly divides m + n is not determined by this fact.
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