| Least Common Multiple (LCM) and Greatest Common Factor (GCF) |
This lesson is intended to show how LCM and GCF are related mathematically. You may think that they are "distant" math concepts. In fact they are closely related by the equation below.
The heart of their "algebraic relationship" is held by the equation...
| LCM (a,b) x GCF (a,b) = ab |
Where:
a and b are numbers
LCM (a,b) is read as "LCM of numbers a and b"
GCF (a,b) is read as "GCF of numbers a and b"
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The equation above tells us that the product of two numbers (a and b) equals the product of their associated LCM and GCF. |
Let's take a look at some examples !
Example 1: Verify that the product of 10 and 15 is equal to the product of LCM(10,15) and GCF (10,15).
Solution:
By quick inspection, GCF(10,15) = 5. For the LCM, notice that the prime factorizations of the two numbers are as follows:
10 = 2¹ x 5¹
15 = 3¹ x 5¹
To find the LCM, multiply together the exponential numbers with the highest powers for "each copy" of the prime number. Each prime number must be represented in the LCM. If the same prime number occurs both from each number, then select the one with the highest power.
LCM(10,15) = 2¹ x 3¹ x 5¹= 30
Verifying the equations, indeed it is true!
LCM(a,b) x GCF(a,b) = ab
30 x 5 = 10 x 15
150 = 150 
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Example 2: Find the GCF and LCM of 24 and 54.
Solution:
If you really think about it, finding the GCF is relatively much easier than LCM. Why not just solve for the GCF using Prime Factorization method and then solve for the LCM using the formula shown above? Sounds like a plan. Let a = 24 and b = 54.
The prime factorizations of the two numbers:
24 = 2 x 2 x 2 x 3 = (23) (31)
54 = 2 x 3 x 3 x 3 = (21) (33)
To find the GCF, multiply together the exponential numbers with the lowest powers for each "common copy" from the listed primes of the given numbers.
GCF(24,54) = 2¹ x 3¹ = 6
Now, to solve for the LCM we use the formula.
Remember our values for a and b: a = 24 and b = 54.
LCM(a,b) x GCF(a,b) = ab
LCM(a,b) x 6 = (24)(54)
LCM(a,b) x 6 = 1,296
| LCM(a,b) |
= |
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= |
216 |
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Therefore, the LCM (24,54) = 216.
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