Determine the order of each of the following elements in the respective products of groups ($D_{19}$ denotes the dihedral group of order $38$ ).

element	product	order
$( 15 \pmod{17}, 10 \pmod{19} )$	$\mathbb{Z}_{17}\times \mathbb{Z}_{19}$
$( 15 \pmod{17}, 10 \pmod{19} )$	$\mathbb{Z}_{17}\times U(19)$
$( 5 \pmod{17}, a^{13} )$	$U( 17 ) \times D_{19}$