The LCM and HCF of two numbers are 1001 and 7 respectively. How many such pairs are possible?

To find the number of pairs of numbers that have a Least Common Multiple (LCM) of 1001 and a Highest Common Factor (HCF) of 7, we need to follow these steps:

Step 1: Understand the relationship between LCM, HCF, and the numbers. The product of two numbers is equal to the product of their LCM and HCF. So, if we let the two numbers be a and b, we have:

a * b = LCM * HCF

Step 2: Substitute the given values into the equation. We know that the LCM is 1001 and the HCF is 7, so we substitute these values into the equation:

a * b = 1001 * 7

Step 3: Simplify the equation. This gives us:

a * b = 7007

Step 4: Find the pairs of factors of 7007. The pairs of factors of 7007 are (1, 7007), (7, 1001), (77, 91), and (143, 49).

Step 5: Check which pairs of factors also have an HCF of 7. The pairs (7, 1001) and (77, 91) both have an HCF of 7.

So, there are 2 pairs of numbers that have an LCM of 1001 and an HCF of 7.