Explanation:

- We need a three-digit number, $x$, which leaves a remainder of 1 when divided by both 8 and 6.

- Mathematically, $x\equiv 1(\mathrm{mod}8)$ and $x\equiv 1(\mathrm{mod}6)$.

- The smallest such number has to satisfy $x-1\equiv 0(\mathrm{mod}24)$ (since 24 is the least common multiple of 8 and 6).

- So, $x=24k+1$ where $k$ is an integer. We need the smallest three-digit $x$.

- Option 1: 121 $\Rightarrow$ Check: 121 - 1 = 120. Does 24 divide 120? Yes.

- Option 2: 119 $\Rightarrow$ 119 - 1 = 118. Does 24 divide 118? No.

- Option 3: 123 $\Rightarrow$ 123 - 1 = 122. Does 24 divide 122? No.

- Option 4: 125 $\Rightarrow$ 125 - 1 = 124. Does 24 divide 124? No.

121 is indeed the correct answer.