Let N, a number greater than 5, be the smaller member of a twin prime pair (e.g. {11,13} {17,19} etc.)

It is easily shown that if m=6 then N mod(m) = 5 - though, of course, 5 is itself the smaller member of a twin prime pair.

More interestingly:
(a) for what other values of m is every possible value of N mod(m) also the smaller member of a twin prime pair?
(b) Explain the phenomenon.

One example is m=12, when every prime pair, the smaller member of which is 11 or more, is worth either 5 or 11, mod 12.

Another is 18, where every such prime pair is worth 5, 11, or 17, mod 18.

Then there is ...?

And the reason is ...?