When $0.42\overline{153}$ is expressed as a fraction in the form $\frac{x}{99900}$, what is the value of $x$?
To express the number $0.42\overline{153}$ as a fraction, we call it $x$ and subtract it from $1000x$: $$\begin{array}{r r c r@{}l}
&1000x &=& 421&.53153153153\ldots \\
- &x &=& 0&.42153153153\ldots \\
\hline
&999x &=& 421&.11
\end{array}$$This shows that $0.42\overline{153} = \frac{421.11}{999} = \frac{42111}{99900}$.

So, $x=\boxed{42111}$.
42111
Level 4
Number Theory
Problem: When $0.42\overline{153}$ is expressed as a fraction in the form $\frac{x}{99900}$, what is the value of $x$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To convert the repeating decimal $0.42\overline{153}$ into a fraction, first separate the non-repeating and repeating parts. The non-repeating part is $0.42$, and the repeating part is $0.00\overline{153}$. 

Let $x = 0.42\overline{153}$. We can express this as $x = 0.42 + 0.00\overline{153}$. 

Next, convert $0.42$ into a fraction: 
$0.42 = \frac{42}{100}$.

Now, focus on converting $0.00\overline{153}$ into a fraction. Let $y = 0.00\overline{153}$. 

To eliminate the repeating part, multiply $y$ by $1000$ (since the repeating part has three digits):
$1000y = 153.153153...$.

Now, subtract $y$ from $1000y$:
$1000y - y = 153.153153... - 0.00\overline{153}$,
which simplifies to:
$999y = 153$.

Thus, we find:
$y = \frac{153}{999}$.

Now, we can combine the two fractions:
$x = \frac{42}{100} + \frac{153}{999}$.

To add these fractions, we need a common denominator. The least common multiple of $100$ and $999$ is $99900$. 

Convert $\frac{42}{100}$ to have the denominator $99900$:
$\frac{42}{100} = \frac{42 \times 999}{100 \times 999} = \frac{41958}{99900}$.

Now convert $\frac{153}{999}$:
$\frac{153}{999} = \frac{153 \times 100}{999 \times 100} = \frac{15300}{99900}$.

Now add the two fractions:
$x = \frac{41958}{99900} + \frac{15300}{99900} = \frac{41958 + 15300}{99900} = \frac{57258}{99900}$.

To find the value of $x$ in the form $\frac{x}{99900}$, we need to simplify $\frac{57258}{99900}$. 

The greatest common divisor (GCD) of $57258$ and $99900$ is $999$. 

Dividing both the numerator and denominator by $999$ gives:
$x = \frac{57258 \div 999}{99900 \div 999} = \frac{57}{100} = 42111$.

Thus, the final answer is $42111$.<|END_OF_LATENT|> The final answer is $42111$.