2.5 Prime factorization and the least common multiple  (Page 4/8)

Find the LCM of $15$ and $18$ using the prime factors method.

Solution

Write each number as a product of primes.
Write each number as a product of primes, matching primes vertically when possible.
Bring down the primes in each column.
Multiply the factors to get the LCM.	$\text{LCM}=2\cdot 3\cdot 3\cdot 5$ The LCM of 15 and 18 is 90.

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Find the LCM of $50$ and $100$ using the prime factors method.

Solution

Write the prime factorization of each number.
Write each number as a product of primes, matching primes vertically when possible.
Bring down the primes in each column.
Multiply the factors to get the LCM.	$\text{LCM}=2\cdot 2\cdot 5\cdot 5$ The LCM of 50 and 100 is 100.

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Key concepts

• Find the prime factorization of a composite number using the tree method.
  1. Find any factor pair of the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime.
  3. If a factor is not prime, write it as the product of a factor pair and continue the process.
  4. Write the composite number as the product of all the circled primes.
• Find the prime factorization of a composite number using the ladder method.
  1. Divide the number by the smallest prime.
  2. Continue dividing by that prime until it no longer divides evenly.
  3. Divide by the next prime until it no longer divides evenly.
  4. Continue until the quotient is a prime.
  5. Write the composite number as the product of all the primes on the sides and top of the ladder.
• Find the LCM using the prime factors method.
  1. Find the prime factorization of each number.
  2. Write each number as a product of primes, matching primes vertically when possible.
  3. Bring down the primes in each column.
  4. Multiply the factors to get the LCM.
• Find the LCM using the prime factors method.
  1. Find the prime factorization of each number.
  2. Write each number as a product of primes, matching primes vertically when possible.
  3. Bring down the primes in each column.
  4. Multiply the factors to get the LCM.

Practice makes perfect

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the factor tree method.

In the following exercises, find the prime factorization of each number using the ladder method.