Fractional part of the number $$\frac{4^{2022}}{15}$$ is equal to

We need to find the fractional part of $$\frac{4^{2022}}{15}$$.

Note that $$4^{2022} = (4^2)^{1011} = 16^{1011}$$.

We need $$4^{2022} \pmod{15}$$.

$$4^1 \equiv 4 \pmod{15}$$

$$4^2 \equiv 16 \equiv 1 \pmod{15}$$

Since $$4^2 \equiv 1 \pmod{15}$$, and $$2022$$ is even:

$$4^{2022} = (4^2)^{1011} \equiv 1^{1011} \equiv 1 \pmod{15}$$

Therefore the fractional part of $$\frac{4^{2022}}{15}$$ is $$\frac{1}{15}$$.

This matches option 4.