If $a$ and $b$ are coprime, prove that $a^2+b^2$ and $a^2b^2$ are coprime

Question

If $a$ and $b$ are coprime, prove that $a^2+b^2$ and $a^2b^2$ are coprime.

Answer:

We have to prove $\gcd(a^2+b^2,a^2b^2)=1$, $\gcd(a,b)=1$. Then $\gcd(a^2,b^2)=1$. Now we have $\gcd(a^2,a^2+b^2)=1$ and $\gcd(b^2,a^2+b^2)=1$ by $\gcd(a,b)=\gcd(p\pm q)$.

Then I am lost.

Answer 1

Let $p$ be a common prime factor of $a^2 + b^2$ and $a^2 b^2$, and recall that for any integer $n$, $p | n \iff p | n^2$. Thus $p | a^2 b^2$ implies $p | ab$, so $p$ must divide both of $a^2 + b^2 \pm 2ab = (a\pm b)^2$; i.e., both $a+b$ and $a-b$ must be multiples of $p$. Thus, both $a$ and $b$ are multiples of $p/2$. If $p$ is odd, then $a$ and $b$ have $p$ as a common factor, so we're done. If $p = 2$, then $p | a^2 b^2$ forces at least one of $a$ and $b$ to be even; as $a^2 + b^2$ is even, both $a$ and $b$ must be even. QED.

Answer 2

Suppose $(a,b)=1$. Bezout's Identity says there are $x,y\in\mathbb{Z}$ so that $$ ax+by=1\tag1 $$ Then $$ 1=(ax+by)^3=\underbrace{\left(ax^3+3bx^2y\right)}_ua^2+\underbrace{\left(3axy^2+by^3\right)}_vb^2\tag2 $$ Rearranging $(2)$, we get $$ u\left(a^2+b^2\right)+(v-u)b^2=1\tag3 $$ and $$ (u-v)a^2+v\left(a^2+b^2\right)=1\tag4 $$ Multiplying $(3)$ and $(4)$ yields $$ \left(\left(a^2+b^2\right)uv+\left(a^2u-b^2v\right)(u-v)\right)\left(a^2+b^2\right)-(u-v)^2a^2b^2=1\tag5 $$ Thus, $\left(a^2+b^2,a^2b^2\right)=1$.