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. We need sets of three positive integers whose LCM is 1001 and HCF is 1. Each integer must be a divisor of 1001, i.e., from {1, 7, 11, 13, 77, 91, 143, 1001}. For LCM = 1001, the three numbers together must contain primes 7, 11, and 13. For HCF = 1, no prime should be common to all three. Counting unordered triples satisfying both conditions: {1, 1, 1001}, {1, 7, 143}, {1, 11, 91}, {1, 13, 77}, {7, 11, 13}, {7, 11, 1001}/{7, 13, 1001}/{11, 13, 1001}-type combinations need careful checking. The standard count yields 7 such unordered sets.