Let $z_1$ and $z_2$ be two complex numbers represented by points on the circle $|z_1| = 1$ and $|z_2| = 2$, respectively. Then
Step-by-Step Solution
Key Concept: Track which remainders are marked in successive rounds using modular arithmetic and the constraint that numbers lie in $[1, 1000]$.
In round 1, integers leaving remainder 1 when divided by 15 are marked; the last is 991, next is $991 + 15 = 1006 > 1000$, so 6 are marked. In round 2, integers leaving remainder 6 are marked; last is 996, next would be $996 + 15 = 1011 > 1000$. In round 3, integers leaving remainder 11 are marked; last is 986, next is $986 + 15 = 1001 > 1000$, which was already marked, so the process terminates.
Correct Answer: 1,2,3