To solve the problem, we observe a cyclic pattern based on the constraints (no two selected numbers differ by 3 or 7). Here's the key reasoning:

Step 1: Identify the cycle length

Since 3 + 7 = 10, the pattern repeats every 10 numbers. For any set of 10 consecutive numbers (e.g., 1-10), we can find the maximum subset where no two elements differ by 3 or 7.

Step 2: Max subset in 1-10

For 1-10, the valid subset is {1,2,3,7} (or equivalent), which has 4 elements. No two elements differ by 3 or 7:

• 1 vs 2/3/7: differences are 1/2/6 (valid)
• 2 vs 3/7: differences are 1/5 (valid)
• 3 vs 7: difference is 4 (valid)

Step 3: Extend to 1000 numbers

1000 = 10 × 100. Each cycle of 10 contributes 4 elements. Thus, total selected numbers = 4 × 100 = 400.

Answer: 400

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