Hands-on math!

Greatest Common Factor (GCF) and
Least Common Multiple (LCM)

GCF and LCM are related concepts, so we will discuss them together:
Greatest Common Factor (GCF)
The greatest common factor (GCF) between 2 numbers is the largest factor that they have in common. One good way to find this number is to write down all the factors of both numbers and then find the biggest one that appears in both lists.

For example, find the GCF of 12 and 8:
    Factors of 12 : 1 , 2 , 3 , 4 , 6 , 12
    Factors of 8   : 1 , 2 , 4 , 8
The highest number that appears in both lists is 4, so 4 is the GCF of 12 and 8.

Now, can you find the GCF of 22 and 33?
    Factors of 22: _________
    Factors of 33: _________
    GCF(22,33) = _____
Least Common Multiple (LCM)
The least common multiple (LCM) of 2 numbers is the smallest number that they both divide evenly into. It is a bigger number than either of the 2 numbers. One good way to find the least common multiple of 2 numbers is to multiply both numbers by 2,3,4,5... and then find the first multiple that appears in both lists.

For example, find the LCM of 6 and 8:
    Multiples of 6: 12 18 24 30 36 42 48 ...
    Multiples of 8: 16 24 32 40 48 56 64 ...
The first number that appears in both lists is 24.
(48 appears also, but it is not 'least'),
so 24 is the LCM of 6 and 8.
Now, can you find the LCM of 10 and 35?
    Multiples of 10: ________________
    Multiples of 35: ________________
    LCM(10,35) = _____
The GCF-LCM Equation
There is an equation that relates GCF and LCM of 2 numbers (a and b):
a x b = LCM(a,b) x GCF(a,b)

Simply put, this says that the greatest common factor of a and b, when multiplied by the least common multiple of a and b, is the product of a and b where a and b are positive integers.
This is a powerful equation! Here's how you use it:

Finding the GCF when you know the LCM
    GCF(a,b) = (a x b) / LCM(a,b)

For our LCM example of 6 and 8 we have:
    GCF (6 , 8) = 6 x 8 / LCM (12 , 8)
    Since the LCM of 6 and 8 is 24, we have:
    GCF (6 , 8) = 6 x 8 / 24 = 48/24 = 2
Finding the LCM when you know the GCF
    LCM(a,b) = (a x b) / GCF(a,b)

For our GCF example of 12 and 8 we have:
    LCM (12 , 8) = 12 x 8 / GCF (12 , 8)
    Since the GCF of 12 and 8 is 4, we have:
    LCM (12 , 8) = 12 x 8 / 4 = 12 x 2 = 24
Finding b when you know a, the GCF of a and b and their LCM:
    b = GCF(a,b) x LCM(a,b) / a

Example: What is b when a = 8, the GCF(a,b) = 4 and their LCM is 24?
    b = 4 x 24 / 8 = 12
Now here's a problem for you:
The GCF of a and b is 12 and their LCM is 240.    a is 48.     What is b?
    b = GCF x LCM / 48 = ___ x ___ / 48           b = _____