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- #1

- Apr 13, 2013

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Which of the following implications are right?

- $a|b^{n} \Rightarrow a|b$
- $a^n|b^n \Rightarrow a|b$
- $a^n|b \Rightarrow a|b$
- $a^3|b^3 \Rightarrow a|b$

That's what I think..

- Wrong.Counterexample: $ 20|10^2 \nRightarrow 20|10$
- Wrong,because $a^n|b^n \Rightarrow b^n=ka^n=(k \cdot a^{n-1}) \cdot a \Rightarrow a|b^n$,and from the first sentence it is wrong..But I have not found a counterexample!
- It is true because $a^n|b \Rightarrow b=ka^n=(k \cdot a^{n-1}) \cdot a \Rightarrow a|b$
- I think it is true,but I don't know how to prove it

Is that what I have tried so far right?