jmc

Testing $i=1,2,3,4,5$ yields that $3^i\equiv 3,2,6,4,5(\mathrm{mod}7)$ respectively, so $i=5.$

Testing $j=1,2,3,4,5$ yields that $5^j\equiv 5,4,6,2,3(\mathrm{mod}7)$ respectively, so $j=5.$

Notice that it is not necessary to test the $i=6$ and $j=6$ cases since we already found the smallest necessary numbers at $i=5$ and $j=5.$

Finally, $ij=5\cdot 5=25\equiv \boxed{1}\mathrm{mod}6.$