My question is too long?

Question

Why would you get the HCF of the given numbers when you multiply the lowest power of each common prime factor?

Answer 1

Because...you're looking for what they have IN COMMON.

For example, if you have 12 = 2 x 2 x 3 and 16 = 2 x 2 x 2 x 2, the biggest factor that they have IN COMMON is the LEAST power of 2 that they BOTH share (2 x 2 = 4).

You have to throw out the 3, because 16 doesn't have 3 as a factor; likewise, you have to throw out the other 2 2's from 16, because the highest power of 2 that is a factor of 16 and also a factor of 12 is 2 x 2 = 4.

For a non-numeric example, suppose you have 3 guys. One speaks English, French, and Spanish; one speaks English and French; and one speaks English and Spanish. The only language that all three have in common is English, so they're forced to use English if all three want to understand.

Or as they say in the military, "You're only as fast as your slowest man."

Answer 2

Highest/Greatest common factor is a number that will divide into each of the given numbers.
eg.
24 = 2 * 2 * 2 * 3
36 = 2 * 2 * 3 * 3
16 = 2 * 2 * 2 * 2

From this we can see that the answer cannot have any prime factors other than 2 or 3 (at best) because no other primes will divide each number.

if we have any more than two 2s, it wont go into 36, similarly, since there is no 3 in 16, we can eexclude that. In short, we need to look at each prime number and pick the lowest number in all of the original numbers.

in this case, that will be two 2s only, so there can be no common factor bigger than 4.

if we drop any of the common primes, by a similar reasoning, the answer will be a common factor, but will not be as high as the one we found above.