Find LCM with prime factorization
Finding the LCM with prime factorization means identifying all the prime factors needed to form each number in the set, then multiplying together the highest power of each prime that appears. For LCM(12, 18, 20): 12 = 2 squared times 3, 18 = 2 times 3 squared, 20 = 2 squared times 5; the LCM uses 2 squared, 3 squared, and 5 = 4 times 9 times 5 = 180. This Grade 7 math skill from Saxon Math, Course 2 provides the most reliable method for finding the Least Common Multiple, especially for three or more numbers.
Key Concepts
Property The LCM of a set of numbers is the product of all the prime factors necessary to form any number in the set.
Examples For 18 and 24: $18 = 2 \cdot 3 \cdot 3$ and $24 = 2 \cdot 2 \cdot 2 \cdot 3$. The LCM needs three 2s and two 3s, so $\operatorname{LCM} = 2^3 \cdot 3^2 = 72$. For 24 and 40: $24 = 2^3 \cdot 3$ and $40 = 2^3 \cdot 5$. The LCM needs three 2s, one 3, and one 5, so $\operatorname{LCM} = 2^3 \cdot 3 \cdot 5 = 120$. For 30 and 75: $30 = 2 \cdot 3 \cdot 5$ and $75 = 3 \cdot 5^2$. The LCM needs one 2, one 3, and two 5s, so $\operatorname{LCM} = 2 \cdot 3 \cdot 5^2 = 150$.
Explanation Break down each number into its prime building blocks, like LEGOs! To find the LCM, you need to gather the maximum number of each prime factor found in any of the original numbers. It is like making sure you have all the necessary pieces to build any of the numbers, then multiplying them together for the smallest super combo.
Common Questions
How do I find the LCM using prime factorization?
Factor each number into primes, then for each prime that appears, take its highest power. Multiply these highest powers together to get the LCM.
What is the LCM of 12 and 18 by prime factorization?
12 = 2 squared times 3; 18 = 2 times 3 squared. Take 2 squared (highest power of 2) and 3 squared (highest power of 3): LCM = 4 times 9 = 36.
Can I use prime factorization for LCM of three numbers?
Yes. Factor all three numbers, then take the highest power of each prime that appears in any of the factorizations. Multiply these together for the LCM.
How is this different from the listing method?
The listing method requires writing out multiples until a common one appears. Prime factorization is more systematic and efficient, especially for large numbers or when finding LCM of multiple numbers.
When do students learn to find LCM by prime factorization?
This method is typically introduced in Grade 6-7. Saxon Math, Course 2 covers it in Chapter 4 as an extension of the basic LCM concept.
Why is LCM by prime factorization reliable for any numbers?
The method works mechanically for any numbers, large or small, with or without common factors. You do not need to recognize factors by inspection or list large sequences of multiples.
What is the difference between finding GCF and LCM by prime factorization?
For GCF, use the LOWEST power of each SHARED prime only. For LCM, use the HIGHEST power of EVERY prime that appears in any factorization. They use the same factorizations but opposite rules.