Do you include zero as a natural number? If one does then the product is zero.
Otherwise, I would use a CAS to do it. See here.
I don't think there is some shortcut for finding product of natural numbers. Zero is not a natural number. It would be better to group numbers in groups of 2s and proceed e.g.,
1 * 2 * 3 * 4* 5 * 6 * 7 * 8 * 9 * 10
= (1 * 10 ) * ( 3 * 9 )*( 4* 5) * (6 * 7) * (8 * 2)
= (10 * 27) *( 20* 42) * 16
= 270 * 840 * 16
etc