Does there exist $a,b,c,d$ such that $\frac{a+b+c+d}{4}$ is an integer?

Question

Let $a,b,c,d$ be defined as such:

$$\{a,b,c,d\} \geq 1,\\ a\neq b\neq c\neq d,\\ a \not\in \{bx,cx,dx\},\\ b \not\in \{ax,cx,dx\},\\ c \not\in \{ax,bx,dx\},\\ d \not\in \{ax,bx,cx\},\\ \{a,b,c,d\} \in \mathbb{Z}$$

In words: $a,b,c,d$ are different positive integers, such that $a, b, c,$ nor $d$ is a multiple of a different variable.

Question:

Does there exist $a,b,c,d$ for $$\\ \frac{a+b+c+d}{4} = I \\ $$ such that $I$ is an integer?

Answer 1

One way is to find two primes that are $1$ mod $4$ and two primes which are $3$ mod $4$.

For instance, $5,7,11,13$.

Answer 2

Just make them all one more than a multiple of 4; for example $(a, b, c, d) = (5, 9, 13, 17)$. This isn't the only way, but it's certainly one way.