Let p be a positive integer such that the unit digit of $$p^{3}$$ is 4. What are the possible unit digits of $$(p+3)^{3}$$?

$$p^3$$ is having unit digit of 4

So, $$p$$ must also be having unit digit of 4

(Since, $$4^3=64$$ is the only case among first 10 natural numbers where unit digit of a perfect cube is 4)

So, $$\left(p+3\right)$$ must have unit digit $$4+3=7$$

Then, unit digit of $$\left(p+3\right)^3$$ will be same as unit digit in case of $$7^3$$ which is 3.       ($$7^3=343$$)

So, unit digit of $$\left(p+3\right)^3$$ is 3.