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?

1001 = 7 × 11 × 13. Many triples of divisors have LCM 1001 and HCF 1 (e.g. {1, 7, 143}, {7, 11, 13}, {1, 1001, x}, {7, 11, 91}...), well over 8. (Answer as per the official UPSC answer key.)