【Question 8】
Given $x=\underbrace{222 \cdots 222}_{K \uparrow 2}$, if $x$ is a multiple of 198, then the smallest value of $K$ that satisfies the condition is $\qquad$
【Analysis and Solution】
$x=\underbrace{222 \cdots 222}_{K \uparrow 2}=\underbrace{111 \cdots 111}_{K \uparrow 1} \times 2$ is a multiple of $198=2 \times 99$;
then we only need $\underbrace{111 \cdots 111}_{K \uparrow 1}$ to be a multiple of 99;
According to the divisibility rule of 99, we know that $\underbrace{11+11+\cdots+11}_{9 \uparrow 11}=99$ is a multiple of 99; so the smallest value of $K$ that meets the condition is $2 \times 9=18$.