Multiples of 32

Consider $\overline{abcde} = 20000 = 32 \cdot 625$ . Then $32 \mid a \alpha \to$ $32 \mid 2 \alpha \to 16 \mid \alpha$ , so $\alpha$ must be a multiple of $16$ , and we let $\alpha = 16k, k\in \mathbb{Z}$ .

Now consider $\overline{abcde} = 12000 = 32 \cdot 375$ . This gives us that $32 \mid 16k + 2\beta \to 8 \mid \beta$ . Thus, it must be the case that $\beta \geq 8$ , since $\beta$ is a positive integer.

Is $\beta = 8$ possible? Yes, it is: let $32 \mid \overline{abcde}$ . This becomes $32 \mid 10000a + 1000b + 100c + 10d + e \to$ $32 \mid 10000a + (992 + 8)b + 100c + 10d + e \to$ $32 \mid 10000a + 32 \cdot 31b + 8 b + 100c + 10d +e \to$ $32 \mid 10000a + 8b + 100c + 10d + e$ .

Thus, setting $(\alpha,\beta,\gamma,\delta,\epsilon) = (10000,8,100,10,1)$ satisfies the condition, and so $\beta = \fbox{8}$ is the smallest positive integer value that satisfies the condition.

Let $\alpha,\beta,\gamma,\delta,\varepsilon$ be a set of integers satisfying the condition. Since $10016$ , $1024$ , $128$ and $32$ are all multiples of $32$ , we must have $\alpha+\delta+6\varepsilon \equiv \beta+2\delta+4\varepsilon \equiv \gamma+2\delta+8\varepsilon \equiv 3\delta+2\varepsilon \equiv 0$ modulo $32$ . Solving these congruences, we deduce that $\alpha \equiv 16\varepsilon \qquad \beta \equiv 8\varepsilon \qquad \gamma \equiv 4\varepsilon \qquad \delta \equiv 10\varepsilon$ modulo $32$ . Thus $\beta$ must be a multiple of $8$ , and hence the smallest possible positive value of $\beta$ is $8$ .

Let's check that $\beta=8$ works. Since $10000\equiv16$ , $1000\equiv8$ and $100\equiv4$ modulo $32$ , we see that $\overline{abcde} \equiv 16a + 8b + 4c + 10d + e$ modulo $32$ , and hence $\overline{abcde}$ is a multiple of $32$ precisely when $16a+8b+4c+10d+e$ is.

$10000a + 1000b + 100c + 10d + e = 32k$ , where $k$ is a positive integer.

$32(312a + 31b + 3c) + 16a + 8b + 4c + 10d + e = 32k$ .

Clearly, the left side of the expression will always be a multiple of $32$ . From the right side of the equation, we get what we are looking for: the minimal values of the coefficients in question. Thus, $\beta = 8$ .

$32 | \overline{abcde} \Rightarrow 32 | (10^4a + 10^3b + 10^2c + 10d + e)$

$\Rightarrow 32 | ((10^4a + 8b + 10^2c + 10d + e) + 32*31b)$

I claim 8 is the minimum possible.

Consider the example: $32 | 81888$ This rules out all possibilities of $\beta = 1,2,...,7$ as the result must be $0 \pmod{8}$

$10000a + 1000b + 100c + 10d + e == 16a + 8b + 4c + 10d + e (mod 32)$

Thus, 8 is the required value

abcde is multiple of 32

We can say 10000a+1000b+100c+10d+e=32k

32 (312a+31b+3c)+16a+8b+4c+10d+e =32k

So 16a+8b+4c+10d+e must be a multiple of 32 so comparing we get beta=8