number theory - Suppose $n=917,4X8,835$ where $X$ is the missing digit. Find possible values of $X$ so that $n$ is divisible by ${2,3,9,25}$

Suppose $n=917,4X8,835$ where X is the missing digit. Find possible
values of $X$ so that $n$ is divisible by each of the integers:
$${2,3,9,25}.$$

I'm a bit confused by the inconsistency by validating if it's possible with the integers.

For example, when checking if it's divisible by $2$ then it just states "$2$ cannot divide $n$, because $2 \nmid 5$ so therefore $2 \nmid n$ for any $X$." Fair enough makes sense.

But then for finding an $X$ that divides $n$ by $3$, then it states "If the sum of all the digits in $n$ is divisible by $3$ then it is divisible by $3$." Which also makes sense, but how come we couldn't do that for $2$ as well? $3 \nmid 5$ as well so if I apply the same logic from $2$ then then there can't be an $X$ right? I don't really understand what's going on. Same with when finding an $X$ that divides it by $9$, shouldn't the last digit need to be dividable by $9$? Then for checking for divisibility by $25$ it just checks the last $2$ digits($35$) is divisible by $25$. Can anyone explain why this is?

In what cases do you check if the "last k digits" are divisible by an integer that is k digits long? What cases do you sum up digits of $n$ and find the $X\in[0,9]$ that makes the sum divisible by the integer?