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Prime Factorization and the Least Common Multiple

Find the prime factorization of a composite number

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.

\[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\]

Condensed — the full section is in OpenStax Prealgebra 2e.

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.

\[\begin{array}{l}10\text{:}10,20,30,40,\ 50,60,70,80,90,100,110,\text{\ldots } \\ 25\text{:}25,\ 50,75,\ 100,125,\text{\ldots }\end{array}\]

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

Try it.

Find the LCM of \(15\) and \(20\) by listing multiples.

Solution

List the first several multiples of \(15\) and of \(20.\) Identify the first common multiple.

\(\begin{array}{l}\text{15:}\ 15,30,45,\ 60,75,90,105,120 \\ \text{20:}\ 20,40,\ 60,80,100,120,140,160\end{array}\)

The smallest number to appear on both lists is \(60,\) so \(60\) is the least common multiple of \(15\) and \(20.\)

Notice that \(120\) is on both lists, too. It is a common multiple, but it is not the least common multiple.

Condensed — the full section is in OpenStax Prealgebra 2e.

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 by listing multiples.
1. List the first several multiples of each number.
2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
3. Look for the smallest number that is common to both lists.
4. This number is 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.

Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.

In the following exercises, identify each as an expression or equation.

In the following exercises, simplify, using the order of operations.

In the following exercises, evaluate each expression.

In the following exercises, translate each phrase into an algebraic expression.

In the following exercises, solve each equation.

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Is \(810\) divisible by \(2,3,5,6,\text{or}\ 10?\)
If you missed this problem, review .

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\(2,3,5,6,10\)
Is \(127\) prime or composite?
If you missed this problem, review .

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prime
Write \(2⋅2⋅2⋅2\) in exponential notation.
If you missed this problem, review .

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\({2}^{4}\)

Find the prime factorization of \(48\) using the factor tree method.

Odhaliť odpoveď

We can start our tree using any factor pair of 48. Let's use 2 and 24. We circle the 2 because it is prime and so that branch is complete.
Now we will factor 24. Let's use 4 and 6.
Neither factor is prime, so we do not circle either. We factor the 4, using 2 and 2. We factor 6, using 2 and 3. We circle the 2s and the 3 since they are prime. Now all of the branches end in a prime.
Write the product of the circled numbers.	\(2⋅2⋅2⋅2⋅3\)
Write in exponential form.	\({2}^{4}⋅3\)

Check this on your own by multiplying all the factors together. The result should be \(48.\)

Find the prime factorization using the factor tree method: \(80\)

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2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 2⁴ ⋅ 5
Find the prime factorization using the factor tree method: \(60\)

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2 ⋅ 2 ⋅ 3 ⋅ 5, or 2² ⋅ 3 ⋅ 5

Find the prime factorization of 84 using the factor tree method.

Odhaliť odpoveď

We start with the factor pair 4 and 21.

Neither factor is prime so we factor them further.

Now the factors are all prime, so we circle them.

Then we write 84 as the product of all circled primes.

\(2⋅2⋅3⋅7\)
\({2}^{2}⋅3⋅7\)

Draw a factor tree of \(84.\)

Find the prime factorization using the factor tree method: \(126\)

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2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 3² ⋅ 7
Find the prime factorization using the factor tree method: \(294\)

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2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 7²

Find the prime factorization of \(120\) using the ladder method.

Odhaliť odpoveď

Divide the number by the smallest prime, which is 2.
Continue dividing by 2 until it no longer divides evenly.
Divide by the next prime, 3.
The quotient, 5, is prime, so the ladder is complete. Write the prime factorization of 120.	\(2⋅2⋅2⋅3⋅5\) \({2}^{3}⋅3⋅5\)

Check this yourself by multiplying the factors. The result should be \(120.\)

Find the prime factorization using the ladder method: \(80\)

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2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 2⁴ ⋅ 5
Find the prime factorization using the ladder method: \(60\)

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2 ⋅ 2 ⋅ 3 ⋅ 5, or 2² ⋅ 3 ⋅ 5

Find the prime factorization of \(48\) using the ladder method.

Odhaliť odpoveď

Divide the number by the smallest prime, 2.
Continue dividing by 2 until it no longer divides evenly.
The quotient, 3, is prime, so the ladder is complete. Write the prime factorization of 48.	\(2⋅2⋅2⋅2⋅3\) \({2}^{4}⋅3\)

Find the prime factorization using the ladder method. \(126\)

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2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 3² ⋅ 7
Find the prime factorization using the ladder method. \(294\)

Odhaliť odpoveď

2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 7²
Find the LCM of \(15\) and \(20\) by listing multiples.

Odhaliť odpoveď

List the first several multiples of \(15\) and of \(20.\) Identify the first common multiple.
\(\begin{array}{l}\text{15:}\ 15,30,45,\ 60,75,90,105,120 \\ \text{20:}\ 20,40,\ 60,80,100,120,140,160\end{array}\)
The smallest number to appear on both lists is \(60,\) so \(60\) is the least common multiple of \(15\) and \(20.\)
Notice that \(120\) is on both lists, too. It is a common multiple, but it is not the least common multiple.
Find the least common multiple (LCM) of the given numbers: \(9\ \text{and}\ 12\)

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36
Find the least common multiple (LCM) of the given numbers: \(18\ \text{and}\ 24\)

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72

Find the LCM of \(15\) and \(18\) using the prime factors method.

Odhaliť odpoveď

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.	\(\text{LCM}=2⋅3⋅3⋅5\) The LCM of 15 and 18 is 90.

Find the LCM using the prime factors method. \(15\ \text{and}\ 20\)

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60
Find the LCM using the prime factors method. \(15\ \text{and}\ 35\)

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105

Find the LCM of \(50\) and \(100\) using the prime factors method.

Odhaliť odpoveď

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.	\(\text{LCM}=2⋅2⋅5⋅5\) The LCM of 50 and 100 is 100.

Find the LCM using the prime factors method: \(55,88\)

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440
Find the LCM using the prime factors method: \(60,72\)

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360
Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)

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40
Grocery shopping Paper plates are sold in packages of \(12\) and party cups come in packs of \(8.\) What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)
Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why?
Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why?
\(24\div 6\)

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24 divided by 6, the quotient of twenty-four and six.
\(50\ge 47\)

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50 is greater than or equal to 47
\(a⋅a⋅a⋅a⋅a\)
\(x⋅x⋅x⋅x⋅x⋅x\)

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x⁶
\(10⋅10⋅10\)
\({10}^{6}\)
\((10+2)⋅5\)
\((30+6)\div 2\)

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18
\(30+6\div 2\)
\({7}^{2}+{5}^{2}\)

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74
\({(7+5)}^{2}\)
\(4+3(10-1)\)

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31

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Prime Factorization and the Least Common Multiple

Find the prime factorization of a composite number
Find the least common multiple (LCM) of two numbers
Find any factor pair of the given number, and use these numbers to create two branches.
If a factor is prime, that branch is complete. Circle the prime.
If a factor is not prime, write it as the product of a factor pair and continue the process.
Write the composite number as the product of all the circled primes.
Divide the number by the smallest prime.
Continue dividing by that prime until it no longer divides evenly.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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