Multiples of 32

What is the smallest possible positive integer value of $\beta,$ such that for some integers $\alpha,$ $\gamma,$ $\delta,$ and $\epsilon$, the following condition is true?

Condition: For every five-digit number $\overline{abcde}$ that is a multiple of $32,$ $\alpha a + \beta b + \gamma c + \delta d + \epsilon e$ is also a multiple of $32.$

Details and assumptions

$\overline{abc}$ means $100a + 10b + 1c$, as opposed to $a \times b \times c$. As an explicit example, for $a=2, b=3, c=4$, $\overline{abc} = 234$ and not $2 \times 3 \times 4 = 24$.