Imagine two kids each sitting in the center of their own tilt-a whirl®, each holding steady a flashlight (or laser) whose beam is parallel to the deck. Each machine rotates and precesses (we'll leave out nutation) at different, real, rates. How do we determine the least number of rotations to alignment, for each kid.
If you prefer another analogy: Saturn and Jupiter orbit with different (real) periods. How many orbits for each until conjunction in precisely the same position in space? Simplify by assuming circular, observer-centric, orbits.
specifically solutions to:
aC1 + bC2 = N
where a, b, N are positive integers of your own choosing and C1, C2 are constant, unique, positive, non-integer(real) numbers where C1 < C2.
I am hoping to avoid rigorous computational methods involving modulus tests.
Hope I clarified the ambiguities.
Thanks, @DonThousand the Bezout identity looks interesting.