5. (10 points) A natural number $A$ is written twice in a row (for example, 12 is written as 1212) to form a new number $B$. If $B$ is a multiple of 2016, then the smallest $A$ is $\qquad$

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5. (10 points) A natural number $A$ is written twice in a row (for example, 12 is written as 1212) to form a new number $B$. If $B$ is a multiple of 2016, then the smallest $A$ is $\qquad$
【Solution】Solution: $2016=2^{5} \times 7 \times 3^{2}$,
Since $B$ is a multiple of 2016, i.e., $B=2016 k$;
Then $A$ is at least a two-digit number, represented as $\overline{\mathrm{ab}}, B=\overline{\mathrm{abab}}=\overline{\mathrm{ab}} \times 101, 101$ has no common divisor with 2016, so $A$ is not the smallest;

Therefore, let's assume $A$ is a three-digit number, represented as $\overline{\mathrm{abc}}$, then $B=\overline{\mathrm{abc}} \times 1001=\overline{\mathrm{abc}} \times 13 \times 11 \times 7$, thus $\overline{\mathrm{abc}} \times 13 \times 11 \times 7=2^{5} \times 7 \times 3^{2} k$,
$\overline{\mathrm{abc}} \times 13 \times 11=2^{5} \times 3^{2} k$,
Since later, $A \times(10001, 100001 \cdots$, are not multiples of 2 and 3),
To make $A$ the smallest, then $A=\overline{\mathrm{abc}}=2^{5} \times 3^{2}=288$;
Answer: The smallest $A$ is 288.
Therefore, the answer is: 288.