How do I calculate the number of times angle X should be added to obtain angle Y?

Question

Suppose I have a angle X=100 and angle Y=60. when we add X 15 times it gives Y. How do I calculate that?

Answer 1

Hint: You want to solve $100x\equiv 60\pmod{360}$. We are using the fact that two angles are coterminal if they differ by an integer multiple of $360^\circ$.

Equivalently (divide through by $20$) you want to solve $5x\equiv 3\pmod{18}$.