The HCF of two numbers is 6 and their LCM is 180. How many such pairs are there in which BOTH numbers are two-digit numbers?

Explanation

1. 01

   Write the numbers as 6a and 6b with a, b co-prime; then $6ab = 180$, so ab = 30.

2. 02

   Co-prime pairs with product 30: (1, 30), (2, 15), (3, 10), (5, 6).

3. 03

   Numbers: (6, 180), (12, 90), (18, 60), (30, 36).

4. 04

   Only (6, 180) has a number that is not two-digit, so 3 pairs remain.