Q. There are n sets of numbers each having only three positive integers with LCM equal to 1001 and HCF equal to 1. What is the value of n?

Explanation : 

1. Factor 1001:
1001 = 7 × 11 × 13.

2. Observe:
• Since LCM of the three numbers is 1001, together they must “cover” each prime 7, 11, 13 at least once.
• Since their HCF is 1, no prime can appear in all three numbers—each prime must be missing from at least one of them.

3. Think of each prime “sticker” (7, 11, 13) that you want to stick onto exactly three “boxes” (the three numbers):
– You can stick it on exactly one box (3 choices), or
– You can stick it on exactly two boxes (3 choices).
You cannot use “zero boxes” (would fail LCM) or “all three boxes” (would fail HCF).
So for each prime there are 3 + 3 = 6 valid ways.

4. Primes act independently.
Number of ordered triples (A, B, C) = 6 (ways for 7) × 6 (ways for 11) × 6 (ways for 13) = 216.

5. Even if we treat (A, B, C) as an unordered set {A,B,C}, we divide 216 by at most 3! = 6, giving at least 36 distinct sets.

6. 36 is clearly more than 8.

Therefore n > 8, so the correct answer is More than 8.