What is the greatest common factor for #84# and for #48#? What about their least common multiple?

The greatest common factor (GCF) is the largest number by which both numbers are divisible, and the least common multiple is the smallest number that is equal to either number multiplied by an integer.

The GCF must divide both numbers, so it must be a product of two or more factors belonging to both numbers at the same time. It turns out that these factors are all prime numbers.

The prime factorization of #84# is #color(green)(2 xx 2 xx 3) xx 7#.
The prime factorization of #48# is #color(green)(2 xx 2) xx 2 xx 2 xx color(green)(3)#.

I've highlighted the common factors. Therefore, the GCF is #2 xx 2 xx 3 = color(blue)(12)#.

The LCM can be found by looking at the above prime factorization and multiplying together the factors unique to each number.

#2 xx 2 " "color(white)(........) xx 3 xx color(green)(7)#
#color(green)(2 xx 2 xx 2 xx 2 xx 3)#

Since the #7# from the prime factors of #84# is not in the prime factorization of #48#...

#2 xx 2 xx 2 xx 2 xx 3 xx 7 = 336#

The LCM of #84# and #48# is #color(blue)(336)#. But is this really the LCM? Let's check...

#336/84 = 4#

#336/48 = 7#

Yeah, it's a multiple of both numbers. Also, neither factor after dividing by #48# or #84# is a multiple or factor of the other factor.

If we know the GCF of two numbers there is a quick way of find their LCM

if the two numbers are #a" & "b#

then

#GCF(a,b)xxLCM(a,b)=axxb#

#GCF(48,84)=12#

as shown by a previous answer

#12 xx LCM=48xx84#

#LCM=(cancel(48)^4xx84)/cancel(12)^1#

#LCM=4xx84=336#