To convert the repeating decimal \(0.82\) (which is interpreted as \(0.828282...\)) to a rational number, you would set \(x\) equal to the repeating decimal:
\[ x = 0.828282... \]
Since the repeating part is 2 digits long (the "82"), you would multiply both sides of the equation by \(100\) (which is \(10^2\), corresponding to the 2 digits). This gives:
\[ 100x = 82.828282... \]
Next, you can set up a second equation:
\[ x = 0.828282... \]
Now, you can subtract the first equation from the second:
\[ 100x - x = 82.828282... - 0.828282... \]
This simplifies to:
\[ 99x = 82 \]
Now you can solve for \(x\):
\[ x = \frac{82}{99} \]
So, the number you would multiply both sides by is \(100\).