# If a product of relatively prime integers is an $n$th power, then each is an $n$th power

Show that if $$n$$, $$a$$, $$b$$, and $$c$$ are positive integers with $$\gcd(a, b) = 1$$ and $$ab = c^n$$, then there are positive integers $$d$$, and $$e$$ such that $$a = d^n$$ and $$b = e^n$$.

I know that (by Bezout) $$\gcd\left(a,b\right) = 1$$ implies $$ax + by = 1$$ for some integers $$x$$ and $$y$$, and also that $$\gcd\left(a^n,b^n\right) = 1$$, but this does not help me.

• Have you considered the prime factorizations of $a$, $b$, and $c$? – Jon Feb 11 '11 at 15:03
• @kira: Please ask questions, don't give orders. Also: please make your titles informative, not sentence fragments. – Arturo Magidin Feb 11 '11 at 16:21
• obvious assuming the fundamental theorem of arithmetic – yoyo Feb 11 '11 at 19:00
• @kira: The title describes your problem exactly; why did you change it back to not having mark-up and to the old phrasing? – Arturo Magidin Feb 11 '11 at 19:46
• Because of this:@kira: Please ask questions, don't give orders. Also: please make your titles informative, not sentence fragments. – Arturo Magidin 3 hours ago – kira Feb 11 '11 at 20:16

$$ab=c^n \overset{\rm Lemma}\Rightarrow c=(a,c)(b,c) \,\Rightarrow\, ab = (a,c)^n(b,c)^n\Rightarrow \dfrac{a}{(b,c)^n}\! = \dfrac{(a,c)^n}b$$ \,\Rightarrow\begin{align} a &= (a,c)^n\\ b &= (b,c)^n\end{align}
where the last inference uses Unique Fractionization [both fractions are irreducible by $$(a,b)\!=\!1$$]
Lemma $$\ \ \color{#c00}{c\mid ab},\,\ \color{#0a0}{(a,b,c)=1}\ \Rightarrow \ c = (a,c)(b,c)\ [=\, (ab,c\color{#0a0}{(a,b,c)}) = (\color{#c00}{ab,c}) = c\,]$$
Remark  Alternatively $$\ (a,c)^n\! = (a^n,c^n) = (a^n,ab) = a(a^n,b) = a$$ and $$\,(b,c)^n = b\,$$ by symmetry where the first equality employs the Freshman's Dream.
As  Weil remarks,  this result can be viewed as the essence of Fermat's method of infinite descent.  It generalizes to rings of algebraic integers but depends upon much deeper results in this more general context, viz. the finiteness of the class number and Dirichlet's unit theorem.