A question on the greatest common divisor of integers and their divisor sum

Question

Suppose that $x, y, z$ are positive integers.

Let $\sigma(x)$ be the sum of the divisors of $x$, and let $\gcd(y, z)$ be the greatest common divisor of $y$ and $z$.

Here is my question:

If the following divisibility constraints hold: $$\gcd(X, Y) = 1$$ $$\gcd(X, \sigma(X)) = 1$$ $$\gcd(Y, \sigma(Y)) = 1$$ $$\gcd(XY, \sigma(XY)) = 1$$ does it follow that either $$\gcd(X, \sigma(Y)) = 1$$ or $$\gcd(Y, \sigma(X)) = 1?$$

Answer 1

From the Wikipedia page on the greatest common divisor, we use the following property:

Property G $$\gcd(a, b \cdot c) = 1 \iff \{\gcd(a, b) = 1 \land \gcd(a, c) = 1\}.$$

In particular, we obtain $$\gcd(XY, \sigma(XY)) = 1 \iff \gcd(XY, \sigma(X)\sigma(Y)) = 1$$ (since $\gcd(X,Y) = 1$ and $\sigma$ is weakly multiplicative) $$\iff \{\gcd(XY, \sigma(X)) = 1 \land \gcd(XY, \sigma(Y)) = 1\}$$ (using Property G) $$\iff \{\gcd(X, \sigma(X)) = 1\} \land \{\gcd(Y, \sigma(X)) = 1\} \land \{\gcd(X, \sigma(Y)) = 1\} \land \{\gcd(Y, \sigma(Y)\} = 1\}$$ (using Property G) $$\iff \{\gcd(Y, \sigma(X)) = 1\} \land \{\gcd(X, \sigma(Y)) = 1\}$$ (since $\gcd(X,\sigma(X)) = 1$ and $\gcd(Y,\sigma(Y)) = 1$).

QED

I hope that everything that I have written out is correct! =)