2.5 Prime Factorization and the Least Common Multiple

Learning objectives

Learning objectives

• Find the prime factorization of a composite number

• Find the least common multiple (LCM) of two numbers

Find the Prime Factorization of a Composite Number

In the previous section, we found the factors of a number. Prime numbers have only two factors, the number $1$ and the prime number itself. Composite numbers have more than two factors, and 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. Finding the prime factorization of a composite number will help you later in this course.

You may want to refer to the following list of prime numbers less than $50$ as you work through this section.

One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.

If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.

We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.

For example, let’s find the prime factorization of $36.$ We can start with any factor pair such as $3$ and $12.$ We write $3$ and $12$ below $36$ with branches connecting them.

The factor $3$ is prime, so we circle it. The factor $12$ is composite, so we need to find its factors. Let’s use $3$ and $4.$ We write these factors on the tree under the $12.$

The factor $3$ is prime, so we circle it. The factor $4$ is composite, and it factors into $2\cdot{}2.$ We write these factors under the $4.$ Since $2$ is prime, we circle both $2s.$

The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.

In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.

Note that we could have started our factor tree with any factor pair of $36.$ We chose $12$ and $3,$ but the same result would have been the same if we had started with $2$ and $18,4$ and $9,\mathrm{or}\,6\,\mathrm{and}\,6.$

Example 2.48

Solution

1. Begin the factor tree. Split $48$ into $2$ and $24$, using $2$ as a prime factor.
2. Factor the remaining branch. Split $24$ into $2$ and $12$, then split $12$ into $2$ and $6$, and finally split $6$ into $2$ and $3$.
3. Collect the prime leaves. The figures show the completed tree with prime factors $2,2,2,2,3$, so the prime factorization is $2\cdot{}2\cdot{}2\cdot{}2\cdot{}3$.
4. Verify the product. Multiply the factors to obtain $2\cdot{}2\cdot{}2\cdot{}2\cdot{}3=48$, confirming the factorization of $48$.

Takeaway: Every leaf of a completed factor tree must be prime, and multiplying the leaves must recover the original number.

Example 2.49

Solution

1. Choose a factor pair. Split $84$ into $4$ and $21$, as shown in the first factor-tree figure.
2. Factor each composite branch. Split $4$ into $2\cdot{}2$ and split $21$ into $3\cdot{}7$.
3. Read the prime factors. The completed tree in the figures has the prime leaves $2,2,3,7$, so the prime factorization is $2\cdot{}2\cdot{}3\cdot{}7$.
4. Verify the result. Multiplying gives $2\cdot{}2\cdot{}3\cdot{}7=84$, confirming that the prime factorization of $84$ is $2^2\cdot{}3\cdot{}7$.

Takeaway: The first split of a factor tree may vary, but every branch must still end with prime factors and the same total product.

Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.

To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for $36,$ we divide $36$ by $2,$ the smallest prime factor of $36.$

To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.

Then we divide by the next prime; so we divide $9$ by $3.$

We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, $3,$ is prime, we stop here.

Do you see why the ladder method is sometimes called stacked division?

The prime factorization is the product of all the primes on the sides and top of the ladder.

Notice that the result is the same as we obtained with the factor tree method.

Example 2.50

Solution

1. Divide by two repeatedly. The ladder figures show $120\mathrm{÷}2=60$, $60\mathrm{÷}2=30$, and $30\mathrm{÷}2=15$, recording three factors of $2$.
2. Move to the next prime. Divide $15$ by $3$ to obtain $5$, recording the factor $3$.
3. Finish with the last prime. Since $5$ is prime, record the factor $5$ and stop at $1$.
4. Write and verify the factorization. The factors are $2,2,2,3,5$, and $2\cdot{}2\cdot{}2\cdot{}3\cdot{}5=120$, so the prime factorization of $120$ is $2^3\cdot{}3\cdot{}5$.

Takeaway: Each division in the ladder contributes one prime factor, and the process ends only when the quotient reaches one.

Example 2.51

Solution

1. Build the ladder. Divide $48$ by $2$ to get $24$, then $24$ by $2$ to get $12$, $12$ by $2$ to get $6$, and $6$ by $2$ to get $3$.
2. Record the factors. The four divisions contribute the prime factors $2,2,2,2$, and the final quotient $3$ contributes one more prime factor.
3. Write the factorization. Since $3$ is prime, the prime factorization is $2\cdot{}2\cdot{}2\cdot{}2\cdot{}3=2^4\cdot{}3$.
4. Verify the product. Multiplying the factors gives $16\cdot{}3=48$, matching the original number and the repeated divisions shown in the figures.

Takeaway: Keep the ladder until the quotient reaches one, because every intermediate quotient may still contain an undiscovered prime factor.

Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of $10$ and $25.$ We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.

We see that $50$ and $100$ appear in both lists. They are common multiples of $10$ and $25.$ We would find more common multiples if we continued the list of multiples for each.

The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of $10$ and $25$ is $50.$

Example 2.52

Solution

1. List the multiples of $15$. Multiplying $15$ by $1,2,3,4,$ and so on gives $15,30,45,60,75$ and continuing multiples.
2. List the multiples of $20$. Multiplying $20$ by $1,2,3,4,$ and so on gives $20,40,60,80,100$ and continuing multiples.
3. Find the first shared value. The first number appearing in both lists is $60$, which appears as the fourth multiple of $15$ and the third multiple of $20$.
4. Check for earlier common values. Every earlier entry in the two lists differs, so no common multiple smaller than $60$ was skipped.
5. State the least common multiple. Since $60$ is the smallest common multiple of $15$ and $20$, the least common multiple is $60$.

Takeaway: The least common multiple is the first number shared by the multiples lists of $15$ and $20$, which is $60$ rather than the later common multiple $120$.

Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of $12$ and $18.$

We start by finding the prime factorization of each number.

Then we write each number as a product of primes, matching primes vertically when possible.

Now we bring down the primes in each column. The LCM is the product of these factors.

Notice that the prime factors of $12$ and the prime factors of $18$ are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that $36$ is the least common multiple.

Example 2.53

Solution

1. Factor both numbers. The figures show $15=3\cdot{}5$ and $18=2\cdot{}3\cdot{}3$.
2. Choose the greatest exponent of each prime. The factor $2$ occurs once, the factor $3$ occurs twice, and the factor $5$ occurs once.
3. Form the least common multiple. Multiply $2\cdot{}3\cdot{}3\cdot{}5$, which equals $90$.
4. Check the result. The figures show the combined prime factors, and $90$ is divisible by both $15$ and $18$, so the least common multiple is $90$.

Takeaway: For the prime-factors method, use each prime to the greatest exponent found in either number rather than multiplying both factorizations.

Example 2.54

Solution

1. Write the prime factorizations. The figures show $50=2\cdot{}5\cdot{}5$ and $100=2\cdot{}2\cdot{}5\cdot{}5$.
2. Select the greatest exponent for each prime. For $2$, select the two copies from $100$; for $5$, both numbers contain two copies, so select two copies of $5$.
3. Multiply the selected factors. Compute $2\cdot{}2\cdot{}5\cdot{}5=100$.
4. Verify the least common multiple. Since $100\mathrm{÷}50=2$ and $100\mathrm{÷}100=1$, both divisions are exact, so the least common multiple of $50$ and $100$ is $100$.

Takeaway: Matching prime factors in an LCM calculation use the greater exponent, and the smaller number may already divide the larger one.

Section 2.5 Exercises

Section 2.5 Exercises

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.

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

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

In the following exercises, find the least common multiple (LCM) using any method.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

1. Find the prime factorization using the factor tree method: $80$

2. Find the prime factorization using the factor tree method: $60$

1. Find the prime factorization using the factor tree method: $126$

2. Find the prime factorization using the factor tree method: $294$

1. Find the prime factorization using the ladder method: $80$

2. Find the prime factorization using the ladder method: $60$

1. Find the prime factorization using the ladder method. $126$

2. Find the prime factorization using the ladder method. $294$

1. $86$

2. $132$

1. $693$

2. $115$

1. $2475$

2. $56$

1. $168$

2. $391$

1. $432$

2. $2160$

1. $150$

2. $525$

1. $36$

2. $350$