Why is the product of the LCM and HCF the same as the product of the original pair of numbers?

It's the way numbers work. Consider 32 and 33. Consecutive integers are relatively prime, that is, their GCF is 1. If two numbers have a GCF of 1, the LCM will be their product.

32 x 33 = 1056

1 (GCF) x 1056 (LCM) = 1056

2 x 528 = 1056

3 x 352 = 1056

4 x 264 = 1056

Notice the pattern. As the GCF increases, the LCM decreases. Consider 32 and 34.

Consecutive even numbers have a GCF of 2. The LCM of 32 and 34 is 544.

32 x 34 = 1088

2 (GCF) x 544 (LCM) = 1088

If you know either the GCF or the LCM of two numbers, you can find the other one without factoring again. The GCF of 28 and 36 is 4. Their product is 1008. Their LCM is 1008 divided by 4, or 252.