### science goals

By the end of this section you will be able to:

- Find prime factorization of a complex number
- Find the least common multiple (LCM) of two numbers

### Be prepared 2.12

Take this readiness test before you start.

Is$810$divisible by$2,3,5,6,\text{Lub}\phantom{\rule{0ex}{0ex}}10?$

If you missed this issue, check it outExample 2.44.

### Be prepared 2.13

Is$127$Prime or Composite?

If you missed this issue, check it outExample 2.47.

To write$2\cdot 2\cdot 2\cdot 2$in exponential notation.

If you missed this issue, check it outExample 2.5.

### Find prime factorization of a complex number

In the previous section, we found the factors of a number. Prime numbers have only two factors, number$1$and prime number itself Composite numbers have more than two divisors, and any composite number can be written as a unique product of prime numbers. It's calledprime factorizationnumber. When we write the prime factorization of a number, we rewrite the number as a product of prime numbers. Finding the prime factorization of a composite number will help you later in this tutorial.

### prime factorization

Factoring a number into prime factors is the product of prime numbers that is equal to that number.

### manipulative mathematics

Doing manipulative math with primes will help you develop a better feel for primes.

You can refer to the following list of primes less than$50$when working with this section.

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

#### Decomposition into prime factors by the factor tree method

One way to find the prime factorization of a number is to create afactor tree. We start by entering a number, and then we write it as the product of two factors. We write the factors under the number and connect them to the number with a small line segment - the "branch" of the factor tree.

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

We continue until all branches end in a prime. When the factor tree is complete, the prime numbers in a circle give us prime factorization.

For example, let's find the prime factorization of z$36.$We can start with any pair of factors, e.g$3$I$12.$We write$3$I$12$pod$36$with connecting branches.

Factor$3$is a prime number, so we circle it. Factor$12$is complex, so we need to find its factors. let's use$3$I$4.$We write these factors in the tree below$12.$

Factor$3$is a prime number, so we circle it. Factor$4$is made up and taken into account$2\xb72.$We sign these factors$4.$From$2$is a prime number, let's circle both$2\text{S}.$

Prime factorization is the product of the circled prime numbers. In general, prime factorization is written in order from smallest to largest.

$$2\cdot 2\cdot 3\cdot 3$$

In those cases where some prime factors repeat, we can write the prime factorization in exponential form.

$$\begin{array}{c}2\cdot 2\cdot 3\cdot 3\\ \\ {2}^{2}\cdot {3}^{2}\end{array}$$

Note that we could have started our factor tree with any pair of factors$36.$We decided$12$I$3,$but the same result would be the same if we started with that$2$I$18,4$I$9,\text{Lub}\phantom{\rule{0ex}{0ex}}6\phantom{\rule{0ex}{0ex}}\text{I}\phantom{\rule{0ex}{0ex}}6.$

### How one

#### Find the prime factorization of a composite number using the tree method.

- Step 1.Find any pair of factors with the given number and use these numbers to create two branches.
- Step 2.If the factor is prime, this branch is complete. Circle the prime number.
- Step 3.If the factor is not prime, write it as a product of a pair of factors and continue.
- step 4Write the composite number as the product of all the prime numbers in the circle.

### Example 2,48

Find prime factorization$48$using the factor tree method.

#### Solution

We can start our tree with any pair of factors 48. Let's use 2 and 24. We circle the number 2 because it is a prime number, so the branch is complete. | |

Now we're factoring 24. Let's take 4 and 6. | |

Since none of the factors are prime, we don't draw a circle either. We circle the numbers 2 and 3 because they are prime numbers. Now all branches end in prime. | |

Write the product of the numbers in a circle. | $2\cdot 2\cdot 2\cdot 2\cdot 3$ |

Save in exponential form. | ${2}^{4}\cdot 3$ |

Check it yourself by multiplying all the factors together. The result should be$48.$

### try it 2,95

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

### try it 2,96

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

### Example 2,49

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

#### Solution

We start with a pair of factors 4 and 21. Since none of the factors are prime, we break them down further. | |

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

Then we'll write 84 as the product of all the circled primes. | $2\cdot 2\cdot 3\cdot 7$ ${2}^{2}\cdot 3\cdot 7$ |

Draw a factor tree$84.$

### try it 2,97

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

### try it 2,98

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

#### Ladder method analysis of prime factors

Theladder methodis another way to find the prime factors of a composite number. This 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 start building the "ladder", divide the given number by its smallest prime factor. For example, to run a ladder$36,$we share$36$von$2,$smallest prime factor$36.$

To add a "step" to the ladder, we keep dividing by the same prime until the division is no longer equal.

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

In this way, we continue to divide the ladder until the quotient is a prime number. Because the quotient$3,$it's Prime, we stop here.

Do you understand why ladder method is sometimes called stack division?

Prime factorization is the product of all the primes on the sides and at the top of the ladder.

$$\begin{array}{c}2\cdot 2\cdot 3\cdot 3\\ \\ {2}^{2}\cdot {3}^{2}\end{array}$$

Note that the result is the same as we got using the factor tree method.

### How one

#### Find the prime factorization of a composite number using the ladder method.

- Step 1.Divide the number by the smallest prime number.
- Step 2.Continue dividing by that prime until dividing is no longer equal.
- Step 3.Divide by the next prime number until division is no longer equal.
- step 4Continue until the quotient is a prime number.
- step 5Write the composite number as the product of all the primes on the sides and at the top of the ladder.

### Example 2,50

Find prime factorization$120$ladder method.

#### Solution

Divide the number by the smallest prime number, which is 2. | |

Continue dividing by 2 until the division is no longer even. | |

Divide by the next prime number, 3. | |

The quotient of 5 is prime, so the ladder is complete. Write the prime factorization of the number 120. | $2\cdot 2\cdot 2\cdot 3\cdot 5$ ${2}^{3}\cdot 3\cdot 5$ |

Check it yourself by multiplying the factors. The result should be$120.$

### try it 2,99

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

### try it 2.100

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

### Example 2.51

Find prime factorization$48$ladder method.

#### Solution

Divide the number by the smallest prime number, 2. | |

Continue dividing by 2 until the division is no longer equal. | |

The quotient of 3 is prime, so the ladder is complete. Write the prime factorization of the number 48. | $2\cdot 2\cdot 2\cdot 2\cdot 3$ ${2}^{4}\cdot 3$ |

### try it 2.101

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

### try it 2.102

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

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

One of the reasons we are interested in multiples and primes is to use these techniques to find the least common multiple of two numbers. This is useful when adding and subtracting fractions with unlike denominators.

#### Multiple method display

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples$10$I$25.$We can list prime multiples of any number. Then we look for multiples that are common to both lists - these are common multiples.

$$\begin{array}{c}10\text{:}10,20,30,40,\phantom{\rule{0ex}{0ex}}50,60,70,80,90,100,110,\text{\u2026}\hfill \\ 25\text{:}25,\phantom{\rule{0ex}{0ex}}50,75,\phantom{\rule{0ex}{0ex}}100,125,\text{\u2026}\hfill \end{array}$$

We see it$50$I$100$appear on both lists. They are common multiples$10$I$25.$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 calledleast common multiple(LCM). So the lowest LCM with$10$I$25$Is$50.$

### How one

#### Find the least common multiple (LCM) of two numbers by listing multiples.

- Step 1.Write the first multiple of each number.
- Step 2.Look for multiples that are common to both lists. If there are no common multiples in the lists, enter additional multiples for each number.
- Step 3.Find the smallest number common to both lists.
- step 4Ten numer to LCM.

### Example 2,52

Find LCM with$15$I$20$listing multiples.

#### Solution

Give the first multiples$15$and from$20.$Find the first common multiple.

$\begin{array}{l}\text{15:}\phantom{\rule{0ex}{0ex}}15,30,45,\phantom{\rule{0ex}{0ex}}60,75,90,105,120\hfill \\ \text{20:}\phantom{\rule{0ex}{0ex}}20,40,\phantom{\rule{0ex}{0ex}}60,80,100,120,140,160\hfill \end{array}$

The smallest number that appears in both lists is$60,$So$60$is the least common multiple$15$I$20.$

notice it$120$is also on both lists. It is a common multiple, but not the least common multiple.

### try it 2.103

Find the least common multiple (LCM) of the given numbers:$9\phantom{\rule{0ex}{0ex}}\text{I}\phantom{\rule{0ex}{0ex}}12$

### try it 2.104

Find the least common multiple (LCM) of the given numbers:$18\phantom{\rule{0ex}{0ex}}\text{I}\phantom{\rule{0ex}{0ex}}24$

#### prime factors method

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

We start by finding the prime factorization of each number.

$$12=2\cdot 2\cdot 3\phantom{\rule{0ex}{0ex}}18=2\cdot 3\cdot 3$$

We then write each number as a product of primes, matching primes vertically whenever possible.

$$\begin{array}{l}12=2\cdot 2\cdot 3\hfill \\ 18=2\cdot \phantom{\rule{0ex}{0ex}}3\cdot 3\end{array}$$

Now we decrease the prime numbers in each column. LCM is the product of these factors.

Note that prime factors$12$and prime factors$18$are included in the LCM. When assigning common primes, each common prime factor is used only once. It ensures it$36$is the least common multiple.

### How one

#### Find the LCM by prime factors method.

- Step 1.Find the prime factorization of any number.
- Step 2.Write each number as a product of primes, matching primes vertically if possible.
- Step 3.Decrease the prime numbers in each column.
- step 4Multiply the factors to get the LCM.

### Example 2,53

Find LCM with$15$I$18$using the prime factor method.

#### Solution

Write each number as a product of prime numbers. | |

Write each number as a product of primes, matching primes vertically if possible. | |

Decrease the prime numbers in each column. | |

Multiply the factors to get the LCM. | $\text{LCM}=2\cdot 3\cdot 3\cdot 5$ LCM 15 and 18 is 90. |

### try it 2.105

Find the LCM by prime factors method.$15\phantom{\rule{0ex}{0ex}}\text{I}\phantom{\rule{0ex}{0ex}}20$

### try it 2.106

Find the LCM by prime factors method.$15\phantom{\rule{0ex}{0ex}}\text{I}\phantom{\rule{0ex}{0ex}}35$

### Example 2,54

Find LCM with$50$I$100$using the prime factor method.

#### Solution

Write down the prime factorization of each number. | |

Write each number as a product of primes, matching primes vertically if possible. | |

Decrease the prime numbers in each column. | |

Multiply the factors to get the LCM. | $\text{LCM}=2\cdot 2\cdot 5\cdot 5$ LCM 50 i 100 to 100. |

### try it 2.107

Find the LCM using the prime factors method:$55,88$

### try it 2.108

Find the LCM using the prime factors method:$60,72$

### media

#### ACCESS TO ADDITIONAL ONLINE RESOURCES

- Example 1: Analysis of the prime factor
- Example 2: Analysis of the prime factor
- Example 3: Prime factorization
- Example 1: Prime factor analysis using cumulative division
- Example 2: Prime factor analysis using cumulative division
- Least common multiple
- Example: Finding the least common multiple from a list of multiples
- Example: Finding the least common multiple using prime factorization

### Section 2.5 Exercises

#### Practice makes champions

**Find prime factorization of a complex number**

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

267.

$86$

268.

$78$

269.

$132$

270.

$455$

271.

$693$

272.

$420$

273.

$115$

274.

$225$

275.

$2475$

276.

1560

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

277.

$56$

278.

$72$

279.

$168$

280.

$252$

281.

$391$

282.

$400$

283.

$432$

284.

$627$

285.

$2160$

286.

$2520$

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

287.

$150$

288.

$180$

289.

$525$

290.

$444$

291.

$36$

292.

$50$

293.

$350$

294.

$144$

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

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

295.

$8,12$

296.

$4,3$

297.

$6,15$

298.

$12,16$

299.

$30,40$

300.

$20,30$

301.

$60,75$

302.

$44,55$

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

303.

$8,12$

304.

$12,16$

305.

$24,30$

306.

$28,40$

307.

$70,84$

308.

$84,90$

Find the Least Common Multiple (LCM) by any method in the following exercises.

309.

$6,21$

310.

$9,15$

311.

$24,30$

312.

$32,40$

#### everyday math

313.

**buy food**Hot dogs are sold in packs of ten, but hot dog buns are sold in packs of eight. What is the smallest number of sausages and rolls you can buy if you want the same number of sausages and rolls? (Hint: it's LCM!)

314.

**buy food**Paper plates are sold in packs$12$and party cups are available in packs of$8.$What is the smallest number of plates and cups you can buy if you want the same number of each? (Hint: it's LCM!)

#### writing exercises

315.

Would you rather find the prime factorization of a composite number using the factor tree method or the ladder method? Why?

316.

Do you prefer to find the LCM by listing multiples or using the method of principal factors? Why?

#### self-examination

ⓐAfter completing the exercises, use this checklist to assess whether you have mastered the objectives of this section.

ⓑOverall, after reading the checklist, do you feel you are well prepared for the next chapter? Why or why not?

## FAQs

### What is 2 and 5 least common factor? ›

LCM of 2 and 5 is **10**.

**What is the LCM of 2 and 5 using prime factorization? ›**

LCM of 2 and 5 by Prime Factorization

Prime factorization of 2 and 5 is **(2) = 2 ^{1} and (5) = 5^{1} respectively**. LCM of 2 and 5 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2

^{1}× 5

^{1}= 10. Hence, the LCM of 2 and 5 by prime factorization is 10.

**What is the easiest way to find the prime factorization of a number? ›**

The simplest algorithm to find the prime factors of a number is to **keep on dividing the original number by prime factors until we get the remainder equal to 1**. For example, prime factorizing the number 30 we get, 30/2 = 15, 15/3 = 5, 5/5 = 1. Since we received the remainder, it cannot be further factorized.

**What is the LCM method example? ›**

LCM denotes the least common factor or multiple of any two or more given integers. For example, **L.C.M of 16 and 20 will be 2 x 2 x 2 x 2 x 5 = 80**, where 80 is the smallest common multiple for numbers 16 and 20.

**What is the LCM of 2 and 8 by prime factorization? ›**

LCM of 2 and 8 by Prime Factorization

Prime factorization of 2 and 8 is (2) = 2^{1} and (2 × 2 × 2) = 2^{3} respectively. LCM of 2 and 8 can be obtained by multiplying prime factors raised to their respective highest power, i.e. **2 ^{3} = 8**. Hence, the LCM of 2 and 8 by prime factorization is 8.

**What is the least common multiple of 2 and 8? ›**

LCM of 2 and 8 is **8**. The smallest number that is a common multiple of a given set of numbers is defined as the least common multiple.

**What is the LCM using prime factorization of 2 and 6? ›**

To find the LCM of 2 and 6 using prime factorization, we will find the prime factors, (2 = 2) and (6 = 2 × 3). LCM of 2 and 6 is the product of prime factors raised to their respective highest exponent among the numbers 2 and 6. ⇒ LCM of 2, 6 = **2 ^{1} × 3^{1} = 6**.

**What are the steps for prime factorization? ›**

Step 1: Divide the given number by its smallest prime factor. Step 2: Divide the quotient obtained in step 1 by its smallest prime factor. Step 3: Continue until the quotient is a 1. Step 4: Write the given number as the product of all the primes that are the divisors of the division.

**What are the three types of LCM? ›**

**Now, let us learn each method with understandable examples.**

- Common multiple method. Learn how to find the LCM by identifying the lowest common multiple from the list of multiplies of the numbers.
- Prime factor method. ...
- Common division method.

**What is the LCM of 8 and 12? ›**

LCM of 8 and 12 is **24**. LCM also known as Least Common multiple or Lowest common multiple is the smallest or the least positive integer that is divisible by the given set of numbers. Consider the example for finding the LCM of 8 and 12. The answer is 24.

### What is the LCM of 40 and 56? ›

LCM of 40 and 56 is **280**. 280 is the smallest/least/first multiple that is common to both 40 and 56. The Least Common Multiple or Lowest Common Multiple known as LCM is the smallest integer that is divisible by the given set of numbers.

**What is prime factor with example? ›**

A prime factor is **a natural number, other than 1, whose only factors are 1 and itself**. The first few prime numbers are actually 2, 3, 5, 7, 11, and so on.

**What is an example of a common prime factor? ›**

Prime Factors: A factor which is a prime number and not a composite number is a prime factor. For example, **2, 3 and 5 are the prime factors of 30**.

**What is the LCM of 8 and 24 using prime factorization? ›**

LCM of 24 and 8 by Prime Factorization

Prime factorization of 24 and 8 is **(2 × 2 × 2 × 3) = 2 ^{3} × 3^{1}** and (2 × 2 × 2) = 2

^{3}respectively. LCM of 24 and 8 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2

^{3}× 3

^{1}= 24. Hence, the LCM of 24 and 8 by prime factorization is 24.

**What is the LCM of 12 and 18 by prime factorization? ›**

**36** is divisible by both 12 and 18. Even 72 is divisible by 12 and 18, however it is not the LCM for 12 and 18. The smaller number than 72 is 36 which is divisible by both 12 and 18. Hence 36 is the Least Common Multiple for 12 and 18.

**What is the LCM of 7 and 12 using prime factorization? ›**

LCM of 7 and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{2} × 3^{1} × 7^{1} = 84. Hence, the LCM of 7 and 12 by prime factorization is **84**.

**What is the LCM of 24 and 30 using prime factorization? ›**

LCM of 24 and 30 can be obtained by **multiplying prime factors raised to their respective highest power**, i.e. 2^{3} × 3^{1} × 5^{1} = 120. Hence, the LCM of 24 and 30 by prime factorization is 120.

**What is the LCM of 14 and 18 using prime factorization? ›**

LCM of 14 and 18 by Prime Factorization

LCM of 14 and 18 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{1} × 3^{2} × 7^{1} = 126. Hence, the LCM of 14 and 18 by prime factorization is **126**.

**What is the LCM of 12 prime factorization? ›**

Prime factorization of 12 = **2 × 2 × 3**.

**What is the LCM of 7 14 and 21 by prime factorization method? ›**

LCM of 7, 14, and 21 by Prime Factorization

Prime factorization of 7, 14, and 21 is **(7) = 7 ^{1}**. LCM of 7, 14, and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2

^{1}× 3

^{1}× 7

^{1}= 42. Hence, the LCM of 7, 14, and 21 by prime factorization is 42.

### What is the LCM of 12 and 14 using prime factorization? ›

LCM of 12 and 14 by Prime Factorization

LCM of 12 and 14 can be obtained by multiplying prime factors raised to their respective highest power, i.e. **2 ^{2} × 3^{1} × 7^{1} = 84**. Hence, the LCM of 12 and 14 by prime factorization is 84.

**What is the prime factorization of 12 using division method? ›**

To find the prime factors, first we will divide the number 12 by its smallest prime factor, that is, 2. Now, divide by the next prime number, that is, 3. So, the prime factorization of 12 = **2 × 2 × 3 or 22 × 3**. This means that 2 and 3 are the prime numbers of 12.

**What is the least common multiple of 8 and 12? ›**

LCM of 8 and 12 is **24**. LCM also known as Least Common multiple or Lowest common multiple is the smallest or the least positive integer that is divisible by the given set of numbers. Consider the example for finding the LCM of 8 and 12. The answer is 24.

**What is the LCM 8 and 9 and 25 using prime factorization? ›**

LCM(8,9,25)= 23×32×52=**1800**.

**What is the LCM of 16 and 12? ›**

LCM of 12 and 16 is **48**. A way of finding the smallest common multiple between any two numbers or more is known as Least Common Multiple (LCM). The numbers (12, 24, 36, 48, 60, 72…) and (16, 32, 48, 64, 80, 96, 112….) are the first few multiples of 12 and 16, respectively.

**What is the LCM of 80 and 120? ›**

LCM of 80 and 120 is **240**. The LCM is the method to find the least common multiple between any two or more numbers.