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 \( 2 \times 3 \times 4 = 24\).

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