Let $$S$$ denote an arithmetic progression whose first term is either 132 or 158, and the common difference is an even integer less than 10. If the $$n^{\text{th}}$$ term of $$S$$ is 174, then the number of possible distinct values of $$n$$ is ____

AP with first term 132 or 158, common difference is an even positive integer $$< 10$$ (so $$d \in \{2,4,6,8\}$$), and the $$n^{\text{th}}$$ term equals 174.

For first term 132: $$132 + (n-1)d = 174 \Rightarrow (n-1)d = 42$$. Divisors of 42 among $$\{2,4,6,8\}$$: 2 ($$n=22$$), 6 ($$n=8$$). 2 values.

For first term 158: $$(n-1)d = 16$$. Divisors among $$\{2,4,6,8\}$$: 2 ($$n=9$$), 4 ($$n=5$$), 8 ($$n=3$$). 3 values.

Distinct $$n$$ values: $$\{3, 5, 8, 9, 22\}$$ — 5 values.