Find the number of sides of a convex polygon if the sum of the measures of its interior angles is: 540, 1080, 1800, 1620, 2340, 3600, 2880, 7560.
We know that the sum of the interior angles of an $n$-gon is $180(n-2)$ degrees. We can set up an equation to find the number of sides, $n$, given the sum of the interior angles, $S$: $$180(n-2) = S.$$We solve for $n$ by dividing both sides of the equation by 180 and then adding 2 to both sides: $$n = \frac{S}{180} + 2.$$Substituting each value of $S$ into this equation yields the number of sides, $n$, for each given sum of the interior angles: \begin{align*}
n &= \frac{540}{180} + 2 = 5,\\
n &= \frac{1080}{180} + 2 = 8,\\
n &= \frac{1800}{180} + 2 = 12,\\
n &= \frac{1620}{180} + 2 = 12,\\
n &= \frac{2340}{180} + 2 = 16,\\
n &= \frac{3600}{180} + 2 = 22,\\
n &= \frac{2880}{180} + 2 = 18,\\
n &= \frac{7560}{180} + 2 = 46.
\end{align*}Therefore, the number of sides of a convex polygon is $\boxed{5, 8, 12, 16, 22, 18, 46}$, for each respective given sum of the interior angles.