Nice uncomplicated approach.
However, even if you didn't know (or understand) this co-prime law, you could consider this:
xyz is definitely a multiple of all three variables. To add only one more to it (xyz + 1) could never give you any common multiple, because the next common multiple must be more than 1 more.
Let's look at x for example:
xyz + 1 = x(yz +1/x)
This can never be a multiple of x because 1/x = 1/prime < 1 (i.e. non integer)
(yz + 1/x) is not a factor
The same argument can be made for each of the other variables too.
Upon a brief inspection, this can be answered in under 5 seconds.

Hence The answer is clearly xyz+1, without even looking at any other options.