If $n$ is odd and $abc=(n-a)(n-b)(n-c),$ then $LCM((n,a),(n,b),(n,c))=n.$

If $n$ is odd and $abc=(n-a)(n-b)(n-c),$ then $LCM((n,a),(n,b),(n,c))=n.$

Prove that if $a,b,c,n\in \mathbb Z^{+},2\not\mid n$ and
$$abc=(n-a)(n-b)(n-c),$$ $x=(n,a),y=(n,b),z=(n,c),$ then $LCM(x,y,z)=n.$
If $n=35,a,b,c=5, 21, 28,$ then
$x=(35,5)=5,y=(35,21)=7,z=(35,28)=7,LCM(x,y,z)=35.$
If $n=945,a,b,c=9, 756, 910,$ then $x,y,z=9, 35, 189,LCM(x,y,z)=945.$
I checked all $n<1000$ and this is always true.
(On the other hand, if $2\mid n$ then $LCM(x,y,z)=n$ or $\dfrac n2.$)