# Relationships between Highest Common Factor and Lowest Common Multiple?

• Mar 13th 2009, 10:44 PM
pepperhu
Relationships between Highest Common Factor and Lowest Common Multiple?
Why is the product of an original pair of numbers same as the product of their Highest Common Factor and Lowest Common Multiple?

For example,
HCF of 12 and 16 = 4
LCM of 12 and 16 = 48

12 x 16 = 192
4 x 48 = 192

Please explain as clearly as possible. Thank You.
• Mar 13th 2009, 11:24 PM
kalagota
Quote:

Originally Posted by pepperhu
Why is the product of an original pair of numbers same as the product of their Highest Common Factor and Lowest Common Multiple?

For example,
HCF of 12 and 16 = 4
LCM of 12 and 16 = 48

12 x 16 = 192
4 x 48 = 192

Please explain as clearly as possible. Thank You.

because it is a theorem..
if x, y are positive numbers, then lcm(x,y) hcf(x,y) = xy..

you may want to read the explanation (proof) here..
Basic Number Theory: LCM/GCD Proof
• Mar 14th 2009, 03:56 AM
Soroban
Hello, pepperhu!

Quote:

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

For example,
HCF of 12 and 16 = 4
LCM of 12 and 16 = 48

12 x 16 = 192
4 x 48 = 192

Please explain as clearly as possible.

Here's an informal explanation.

The HCF contains the "overlap" of their factors.

The LCM contains their factors minus their "overlap".

Together (multiplied), they contain all their factors.