Is the positive integer n a multiple of 3?

There's simply no relationship between divisibility by 2 (or by 4) and divisibility by 3, so there's no reason either Statement would even be useful here. Of course we can check numbers to confirm: using both Statements, n can be 3, which is divisible by 3, or n can be 7, which is not divisible by 3. So the answer is E.

This is a question based on the concept of consecutive integers.
Any set of 3 consecutive integers will always have a multiple of 3. As such, from the data given, we need to determine if the set of integers is in such a way that ‘n’ is a multiple of 3.

From statement I alone, (n-1) is a multiple of 2. In other words, (n-1) is even.
If (n-1) = 2, n = 3. Is n a multiple of 3? YES.
If (n-1) = 4, n = 5. Is n a multiple of 3? NO.
Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, (n+1) is a multiple of 4.
If (n+1) = 4, n = 3. Is n a multiple of 3? YES.
If (n+1) = 12, n = 11. Is n a multiple of 3? NO.
Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:
From statement I, (n-1) is even. From statement II, (n+1) is a multiple of 4.
If (n-1) = 2, n = 3 and (n+1) = 4. Both conditions are met. Is n a multiple of 3? YES.
If (n-1) = 10, n = 11 and (n+1) = 12. Both conditions are met. Is n a multiple of 3? NO.

The combination of statements is insufficient to give us a definite YES or NO. Answer option C can be eliminated.
The correct answer option is E.

Hope that helps!

Statement 1 => n-1 = 2k
n= 2k+1

But we need to check if n = 3p
Insufficient

Statement 2 => n+1 = 4m
n= 4m-1

But we need to check if n = 3p
Insufficient

Combining
n = 2k+1 = 3,5,7,9,11,13,15
n = 4m-1 = 3,7,11,15

Taking common set
3,7,11,15
Some are, some aren't multiples of 3

Insufficient.
E