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 $[1, 1000]$.
Step 1: Analyze Round 1 Marking
Explanation: In the first round, integers up to 1000 that leave a remainder of 1 when divided by 15 are identified and marked. We determine the largest such integer within the range and confirm its boundary.
Math:
The integers marked are of the form $15k + 1$.
The last such integer less than or equal to 1000 is $991$.
The next potential integer in this sequence would be $991 + 15 = 1006$.
Since $1006 > 1000$, this integer is outside the considered range.
A total of 6 integers are marked in this round.
Step 2: Analyze Round 2 Marking
Explanation: In the second round, integers up to 1000 that leave a remainder of 6 when divided by 15 are identified and marked. Similar to Round 1, we determine the largest such integer and the subsequent one.
Math:
The integers marked are of the form $15k + 6$.
The last such integer less than or equal to 1000 is $996$.
The next potential integer in this sequence would be $996 + 15 = 1011$.
Since $1011 > 1000$, this integer is outside the considered range.
Step 3: Analyze Round 3 Marking and Process Termination
Explanation: In the third round, integers up to 1000 that leave a remainder of 11 when divided by 15 are identified. The process's termination condition is determined by the next potential integer in this sequence.
Math:
The integers marked are of the form $15k + 11$.
The last integer satisfying this condition and less than or equal to 1000 is $986$.
The next potential integer in this sequence would be $986 + 15 = 1001$.
The solution explicitly states that $1001 > 1000$ and that this integer "was already marked". This condition indicates that the process for marking numbers terminates.
Final Step: Conclusion of the Marking Process
The provided solution describes a process of marking integers up to 1000 based on their remainders when divided by 15. The process concludes after Round 3 because the next number that would follow the pattern (1001) is both outside the defined range (greater than 1000) and stated to have been "already marked". The final conclusion derived from the provided solution is that the described marking process terminates. This conclusion is not a numerical value and therefore cannot be matched with the options given for the complex number problem.
Correct Answer: 1,2,3