Lesson 5: Prime Factorization and the Least Common Multiple

New Concept

This lesson introduces prime factorization, breaking a number into its prime building blocks. You'll master two methods—factor trees and ladders—and apply this skill to find the Least Common Multiple (LCM), a key tool for working with fractions.

What’s next

First, you'll master prime factorization with interactive examples. Then, you'll apply this skill to find the LCM in a series of practice cards and problems.

Property

The prime factorization of a number is the product of prime numbers that equals the number. Every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes.

Examples

• The prime factorization of 30 is $2 \cdot 3 \cdot 5$, which cannot be simplified further in exponential form.
• For the number 77, the prime factorization is simply $7 \cdot 11$, since both are prime numbers.
• The prime factorization for 84 is $2 \cdot 2 \cdot 3 \cdot 7$, or $2^2 \cdot 3 \cdot 7$.

Explanation

Think of prime factorization as finding a number's unique recipe. Each composite number is built from a special combination of prime number ingredients that, when multiplied together, give you back the original number.

Property

To find the prime factorization of a composite number using the tree method:
Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
Step 2. If a factor is prime, that branch is complete. Circle the prime.
Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
Step 4. Write the composite number as the product of all the circled primes.

Examples

• To factor 36, start with $6 \cdot 6$. Break the first 6 into $2 \cdot 3$ and circle both primes. Break the second 6 into $2 \cdot 3$ and circle both primes. The factorization is $2^2 \cdot 3^2$.
• For 28, branch into $4 \cdot 7$. Circle the prime 7. Branch 4 into $2 \cdot 2$ and circle both 2s. The result is $2^2 \cdot 7$.
• To factor 60, you can start with $6 \cdot 10$. Branch 6 into $2 \cdot 3$ and 10 into $2 \cdot 5$. All are prime, so the factorization is $2^2 \cdot 3 \cdot 5$.

Explanation

This visual method lets you branch out from a number, splitting it into pairs of factors. You keep branching until every path ends in a prime number, which you circle like a bud on the tree.

Property

To find the prime factorization of a composite number using the ladder method:
Step 1. Divide the number by the smallest prime.
Step 2. Continue dividing by that prime until it no longer divides evenly.
Step 3. Divide by the next prime until it no longer divides evenly.
Step 4. Continue until the quotient is a prime.
Step 5. Write the composite number as the product of all the primes on the sides and top of the ladder.

Examples

• For 60, divide by 2 to get 30, by 2 again for 15, and then divide 15 by 3 to get 5. The primes are $2, 2, 3, 5$, so the factorization is $2^2 \cdot 3 \cdot 5$.
• To factor 99, divide by its smallest prime factor, 3, to get 33. Divide by 3 again to get 11. The primes are $3, 3, 11$, giving $3^2 \cdot 11$.
• For 200, divide by 2 to get 100, by 2 for 50, by 2 for 25. Now divide by 5 for 5. The factorization is $2^3 \cdot 5^2$.

Explanation

This method is like a step-by-step division game, also known as stacked division. You repeatedly divide a number by its smallest prime factors until you're left with a prime number at the top of the “ladder.”

Property

The smallest number that is a multiple of two numbers is called the least common multiple (LCM).
To find the least common multiple (LCM) of two numbers by listing multiples:
Step 1. List the first several multiples of each number.
Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
Step 3. Look for the smallest number that is common to both lists.
Step 4. This number is the LCM.

Examples

• For 10 and 14: Multiples of 10 are $10, 20, 30, 40, ...$ and multiples of 14 are $14, 28, 42, ...$. The first common number is 70, so the $\operatorname{LCM}(10, 14) = 70$.
• For 9 and 15: Multiples of 9 are $9, 18, 27, 36, 45, ...$ and multiples of 15 are $15, 30, 45, ...$. The $\operatorname{LCM}(9, 15) = 45$.
• For 7 and 14: Multiples of 7 are $7, 14, 21, ...$ and multiples of 14 are $14, 28, ...$. The first one in common is 14, so the $\operatorname{LCM}(7, 14) = 14$.

Explanation

The LCM is the smallest positive number that is a multiple of two or more numbers. It's the first “meeting point” if you imagine listing out the multiples of each number in a sequence.

Property

To find the LCM using the prime factors method:
Step 1. Find the prime factorization of each number.
Step 2. Write each number as a product of primes, matching primes vertically when possible.
Step 3. Bring down the primes in each column.
Step 4. Multiply the factors to get the LCM.

Examples

• For 28 and 40: $28 = 2^2 \cdot 7$ and $40 = 2^3 \cdot 5$. The LCM needs the highest powers of all primes involved: $2^3 \cdot 5 \cdot 7 = 280$.
• For 20 and 30: $20 = 2^2 \cdot 5$ and $30 = 2 \cdot 3 \cdot 5$. Line up the factors and bring down the highest power of each prime: $2^2 \cdot 3 \cdot 5 = 60$.
• For 50 and 75: $50 = 2 \cdot 5^2$ and $75 = 3 \cdot 5^2$. The LCM is built from $2^1$, $3^1$, and $5^2$. So, $\operatorname{LCM}(50, 75) = 2 \cdot 3 \cdot 5^2 = 150$.

Explanation

This powerful method builds the LCM by taking every prime factor from all numbers at its highest power. This guarantees the result is divisible by all original numbers, making it the least common multiple.