To multiply \(-22\) by \(-\frac{56}{209}\), we first note that the product of two negative numbers is positive. Thus, we can write:
\[ -22 \times -\frac{56}{209} = 22 \times \frac{56}{209} \]
Next, we can express \(22\) as a fraction:
\[ 22 = \frac{22}{1} \]
Now we can multiply the fractions:
\[ \frac{22}{1} \times \frac{56}{209} = \frac{22 \times 56}{1 \times 209} = \frac{1232}{209} \]
Next, we can simplify \(\frac{1232}{209}\) if possible. First, we find the GCD (greatest common divisor) of 1232 and 209.
After performing the calculation, we find that \(209\) divides evenly into \(1232\):
\[ 1232 \div 209 = 5.9 \quad \text{(this shows they don't share factors)} \]
So, the fraction \(\frac{1232}{209}\) is already in its simplest form.
To convert \(\frac{1232}{209}\) to a mixed number:
\[ 1232 \div 209 = 5 \quad \text{(with a remainder of 117)} \] Thus: \[ \frac{1232}{209} = 5 + \frac{117}{209} \]
However, the answer options appear to refer to simple fractions. Since \(1232\) is much greater than \(209\), let's check back if any of the provided options simplify correctly to facilitate the comparison.
Revisiting the provided options:
- \(\frac{56}{19}\)
- \(\frac{2299}{28}\)
- \(-\frac{17}{209}\)
- \(\frac{112}{19}\)
The integer associated with \(22 \times \frac{56}{209}\) computes to \(5.9 \text{ not equivalent to any provided answers}\).
Our only feasible conclusion is to express as:
- Separated as \( \frac{1232}{209} = 5.9\) meaning none match the context.
Please check again if there is any mistake in options or assist in further calculations!
So, the answer simplifies to \( \frac{1232}{209} = \frac{1232}{209} \) thus none of the above options fit within the context of rational number properties satisfactory for the multiplication displayed.
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