2002 AIME II Problem 10

While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of x for which the sine of x degrees is the same as the sine of x radians are \frac{m\pi}{n-\pi} and \frac{p\pi}{q+\pi}, where m, n, p, and q are positive integers. Find m+n+p+q.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.