I was studying elementary number theory and came up with a doubt regarding a statement as follows:
"x,m,n belongs to set of postive integers.
gcd(x,mn)=1 if and only if gcd(x,m)=gcd(x,n)=1"
In the book it is written that the above statement is true if gcd(m,n)=1
Isn't the statement true for all m and n belonging to the set of positive integers? If not,then why?
Note that $ \text{gcd}(m,n) = 1 $ iff there exist integers $y,z$ such that $my + nz = 1$.
$\text{gcd}(x, mn) = 1 $ implies there exist integers $y,z$ such that $xy + mnz = 1$. But then $xy + m(nz) = 1$ and $xy + n(mz) = 1$, so $ \text{gcd}(m,x) = 1 $ and $ \text{gcd}(n,x) = 1 $.
Conversely, if $y_1 m + z_1 x = 1$ and $y_2 n + z_2 x = 1 $, then $$ (y_1 m + z_1 x = 1)( y_2 n + z_2 x = 1) = (y_1y_2)mn + (y_1 z_2 + z_1 y_2 + z_1 z_2 x)x = 1, $$ i.e $\text{gcd}(mn,x) = 1$
