The number of elements in the set $$\{n \in \mathbb{N} : 10 \leq n \leq 100$$ and $$3^n - 3$$ is a multiple of $$7\}$$ is _____.

We need $$3^n - 3 \equiv 0 \pmod{7}$$, i.e., $$3^n \equiv 3 \pmod{7}$$, i.e., $$3^{n-1} \equiv 1 \pmod{7}$$.

The order of 3 modulo 7: $$3^1 = 3$$, $$3^2 = 9 \equiv 2$$, $$3^3 = 27 \equiv 6$$, $$3^4 = 81 \equiv 4$$, $$3^5 = 243 \equiv 5$$, $$3^6 = 729 \equiv 1 \pmod{7}$$.

So $$3^{n-1} \equiv 1 \pmod{7}$$ when $$6 | (n-1)$$, i.e., $$n \equiv 1 \pmod{6}$$.

Values of $$n$$ in $$[10, 100]$$ with $$n \equiv 1 \pmod{6}$$:

$$n = 13$$, $$19$$, $$25$$, $$31$$, $$37$$, $$43$$, $$49$$, $$55$$, $$61$$, $$67$$, $$73$$, $$79$$, $$85$$, $$91$$, $$97$$

Count: $$\frac{97 - 13}{6} + 1 = \frac{84}{6} + 1 = 14 + 1 = 15$$

The answer is $$\mathbf{15}$$.