We need the fractional part of $\left\{\frac{3^{200}}{8}\right\}.$

The fractional part of a number $x$ is $\{x\}=x-\lfloor x\rfloor.$ So for a rational number with denominator $8$, we just need the remainder of the numerator modulo $8$.

Compute $3^{200} \pmod{8}$.

First note: $3^2=9 \equiv 1 \pmod{8}.$

Therefore, $3^{200}=(3^2)^{100} \equiv 1^{100}=1 \pmod{8}.$

So we can write $3^{200}=8k+1$ for some integer $k$.

Hence, $\frac{3^{200}}{8}=k+\frac{1}{8}.$

Therefore, the fractional part is $\left\{\frac{3^{200}}{8}\right\}=\frac{1}{8}.$

Comparing with the options:

• A: $\frac{5}{8}$
• B: $\frac{7}{8}$
• C: $\frac{1}{8}$
• D: $\frac{3}{8}$

The correct option is C.