The angles of quadrilateral $ABCD$ satisfy $\angle A=2\angle B=3\angle C=5\angle D$. What is the degree measure of $\angle A$, rounded to the nearest whole number?

1. Let $\angle A=2\angle B=3\angle C=5\angle D=k$, so $\angle B=\frac k2$, $\angle C=\frac k3$, $\angle D=\frac k5$.
2. $k\left(1+\frac12+\frac13+\frac15\right)=\frac{61}{30}k=360^\circ$, so $k=\dfrac{10800}{61}\approx177.05^\circ$.
3. Rounded, $\angle A\approx177^\circ$.