For a proof in my textbook it is asserted that if $d=(n,a), n=db, a=dc$ for suitable $b,c\in\mathbb{Z}$ then $(b,c)=1$. I am not very good with number theory and this boggles me a bit. I am sure it is not so difficult to realize but I am stumped right now. Could someone please help me out?
Could someone please help me with this proof?     Suppose that $a, b \in N$, and $d = \gcd(a, b)$. Since $d$ divides $a$, we have $a = de$ for some integer $e,$ and similarly $b = df$ for some integer $f$. Prove that $\gcd(e, f) = 1$.   I understand why it works. Since d is all the common factors of $a$ and $b, e$ and $f$ had no common factors, therefore the $\gcd(e,f) = 1$. But how do I prove this?  Thanks in advance.
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