In a certain code, PLUS is written as $6275$ and ATOM is written as $4813$. How is PALM written in that code?

This is a case of direct substitution coding where each letter is assigned a specific digit:
From **PLUS $\to$ $6275$ **:

* $\text{P}=6$
* $\text{L}=2$
* $\text{U}=7$
* $\text{S}=5$

From **ATOM $\to$ $4813$ **:

* $\text{A}=4$
* $\text{T}=8$
* $\text{O}=1$
* $\text{M}=3$

Now, matching the letters for PALM:

* $\text{P}=6$
* $\text{A}=4$
* $\text{L}=2$
* $\text{M}=3$

Thus, PALM is coded as $6423$.