The authors say (1) that division by 2 is much easier (though not trivial) and (2) that their methods generalize to enable you to divide by any other odd number

The authors say (1) that division by 2 is much easier (though not trivial) and (2) that their methods generalize to enable you to divide by any other odd number (and hence, together with division by 2, to divide by any positive integer). So what's special about 3 is that it's the smallest tricky case and that once you've done it you've effectively done everything.