To find the least common multiple (LCM) of $$12$$ and $$18$$ using the prime factors method:
1. Write each as a product of primes: $$12 = 2 \cdot 2 \cdot 3$$ and $$18 = 2 \cdot 3 \cdot 3$$.
2. List the primes and match them vertically: the first $$2$$ and the first $$3$$ are common to both.
3. Bring down the numbers from each column: $$2$$, $$2$$, $$3$$, and $$3$$.
4. Multiply the factors: $$2 \cdot 2 \cdot 3 \cdot 3 = 36$$.
Therefore, the least common multiple of $$12$$ and $$18$$ is $$36$$.

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To find the least common multiple (LCM) using the Prime Factors Method, perform the following steps: Step 1. Write each number as a product of prime factors. Step 2. List the prime factors of each number, matching identical primes vertically in columns whenever possible. Step 3. Bring down one prime from each column. If a column contains the same prime at the top and bottom, bring it down only once. If a column contains only one prime, bring that number down. Step 4. Multiply all the factors brought down from the columns to obtain the LCM. By matching up the common primes, you ensure that each shared prime factor is used only once, which guarantees the resulting product is the least common multiple.

Finding the Least Common Multiple Using the Prime Factors Method

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Example of Finding the Least Common Multiple Using the Prime Factors Method

Find the least common multiple (LCM) of $$9$$ and $$12$$ using the Prime Factors Method. Step 1. Write the prime factorization for each number: $$9 = 3 \cdot 3$$ and $$12 = 2 \cdot 2 \cdot 3$$. Step 2. Align matching prime factors vertically: $$9 = \quad \quad 3 \cdot 3$$ $$12 = 2 \cdot 2 \cdot 3$$ Step 3. Bring down one number from each column: $$2$$, $$2$$, $$3$$, and $$3$$. Step 4. Multiply the resulting factors to find the LCM: $$	ext{LCM} = 2 \cdot 2 \cdot 3 \cdot 3 = 36$$. The least common multiple of $$9$$ and $$12$$ is $$36$$.

Finding the LCM of 9 and 12 Using Prime Factors

Find the least common multiple (LCM) of $$18$$ and $$24$$ using the Prime Factors Method. Step 1. List the prime factorization for both numbers: $$18 = 2 \cdot 3 \cdot 3$$ and $$24 = 2 \cdot 2 \cdot 2 \cdot 3$$. Step 2. Match the prime factors vertically in columns: $$18 = \quad 2 \cdot 3 \cdot 3$$ $$24 = 2 \cdot 2 \cdot 2 \cdot 3$$ Step 3. Bring down one factor from each column: $$2$$, $$2$$, $$2$$, $$3$$, and $$3$$. Step 4. Multiply all the factors together to compute the LCM: $$	ext{LCM} = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 72$$. The least common multiple of $$18$$ and $$24$$ is $$72$$.

Finding the LCM of 18 and 24 Using Prime Factors

Find the least common multiple (LCM) of $$12$$ and $$18$$ using the Prime Factors Method. Step 1. Write each as a product of primes: $$12 = 2 \cdot 2 \cdot 3$$ and $$18 = 2 \cdot 3 \cdot 3$$. Step 2. List the primes and match them vertically in columns: $$12 = 2 \cdot 2 \cdot 3$$ $$18 = 2 \cdot \quad 3 \cdot 3$$ Step 3. Bring down one prime factor from each column to get $$2$$, $$2$$, $$3$$, and $$3$$. Step 4. Multiply these factors together: $$	ext{LCM} = 2 \cdot 2 \cdot 3 \cdot 3 = 36$$. The prime factors of $$12$$ ($$2 \cdot 2 \cdot 3$$) and the prime factors of $$18$$ ($$2 \cdot 3 \cdot 3$$) are all included in the LCM ($$2 \cdot 2 \cdot 3 \cdot 3$$). By matching up the common primes, each shared prime factor is used only once, ensuring $$36$$ is the least common multiple.

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