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Thread: Number Theory

  1. #1
    Junior Member
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    Question Number Theory

    Prove or Disprove the following statements

    If a,b in Z bot greater than 0, and a^2|b^3, than a|b
    If a in Z, a>0, p is prime, and p^4|a^2, then p^2|a.
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  2. #2
    Super Member PaulRS's Avatar
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    1. If $\displaystyle b=n^{2}$ and $\displaystyle a=n^3$ for any integer $\displaystyle n\neq{0}$ and $\displaystyle n\neq{\pm{1}}$then $\displaystyle a^2=b^3$ so $\displaystyle a^2|b^3$ but obviously $\displaystyle a|b$ doesn't hold

    2. This is true.By Euclid's Lemma p|a. Suppose $\displaystyle
    p^s
    $ is the max. power of p dividing $\displaystyle a$, since $\displaystyle p^4|a^2$ it follows that $\displaystyle 2s\geq{4}$ so $\displaystyle s\geq{2}$, therfore$\displaystyle p^2$ divides $\displaystyle a$(since s is greater or equal than 2)
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  3. #3
    Newbie Catherine Morland's Avatar
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    Counterexample for first problem: a = 8, b = 4.
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