Number Theory - Powers of (number) modulo (number)

Question

Why does 3^n modulo 17 (n=0, 1, 2 ...) result in all numbers between 1 and 16 whereas 3^n modulo 26 (n=0, 1, 2 ...) only result in 1, 3, & 9?
3^0 mod 17 = 1
3^1 mod 17 = 3
3^2 mod 17 = 9
3^3 mod 17 = 10
...
3^15 mod 17 = 6
3^n mod 17 with n = 0 -> 15 will have different results, and combined result in all numbers between 1 and 16.
However,
3^0 mod 26
3^1 mod 26
3^2 mod 26
...
3^25 mod 26
only results in 1, 3, & 9.

Answer 1

The groups of units (= invertible elements) of $\mathbf Z/17\mathbf Z$ and of $\mathbf Z/26\mathbf Z$ are both cyclic, of orders $16$ and $12$ respectively.

However, this does not mean that any element $\ne 1$ is a generator, even if the ring is a field: in $\mathbf Z/17\mathbf Z$, $2$ has order $8$, $4$ has order $4$.