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- #1

- Apr 13, 2013

- 3,718

I am given the following exercise:

If $p$ is a prime and $(a,p^2)=p$,$(b,p^3)=p^2$,find $(ab,p^4)$.

$p|a \Rightarrow a=k \cdot p , k\in \mathbb{Z}$

$p^2|b \Rightarrow b=l \cdot p^2 , l \in \mathbb{Z}$

Let $(ab,p^4)=d>1$,then $d$ will have a prime divisor, $q$

$q|d , d|ab \Rightarrow q|ab \Rightarrow q|a \text{ or } q|b$

Also, $d|p^4 \Rightarrow q|p^4 \Rightarrow q=p$

Is it right so far?And how can I continue??