m81273 Prime Factorization and the Least Common Multiple By the end of this section, you will be able to: Find the prime factorization of a composite number Find the least common multiple (LCM) of two numbers f844b6a5-7d09-4876-b9dc-aa70a9d3a4af Before you get started, take this readiness quiz. Is 810 divisible by 2,3,5,6,or10?If you missed this problem, review . 2,3,5,6,10 Is 127 prime or composite?If you missed this problem, review . prime Write 2⋅2⋅2⋅2 in exponential notation.If you missed this problem, review . 24

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. The prime factorization of a number is the product of prime numbers that equals the number. Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better sense of prime numbers.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

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 2·2. 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. 2⋅2⋅3⋅3 In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form. 2⋅2⋅3⋅322⋅32 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,or6and6. 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. Find the prime factorization of 48 using the factor tree method. 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. 24⋅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 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 2⁴ ⋅ 5 Find the prime factorization using the factor tree method: 60 2 ⋅ 2 ⋅ 3 ⋅ 5, or 2² ⋅ 3 ⋅ 5 Find the prime factorization of 84 using the factor tree method. 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 22⋅3⋅7 Draw a factor tree of 84. Find the prime factorization using the factor tree method: 126 2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 3² ⋅ 7 Find the prime factorization using the factor tree method: 294 2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 7²

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. 2⋅2⋅3⋅322⋅32 Notice that the result is the same as we obtained with the factor tree method. Divide the number by the smallest prime. Continue dividing by that prime until it no longer divides evenly. Divide by the next prime until it no longer divides evenly. Continue until the quotient is a prime. Write the composite number as the product of all the primes on the sides and top of the ladder. Find the prime factorization of 120 using the ladder method. 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 23⋅3⋅5 Check this yourself by multiplying the factors. The result should be 120. Find the prime factorization using the ladder method: 80 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 2⁴ ⋅ 5 Find the prime factorization using the ladder method: 60 2 ⋅ 2 ⋅ 3 ⋅ 5, or 2² ⋅ 3 ⋅ 5 Find the prime factorization of 48 using the ladder method. 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 24⋅3 Find the prime factorization using the ladder method. 126 2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 3² ⋅ 7 Find the prime factorization using the ladder method. 294 2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 7²

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. 10:10,20,30,40,50,60,70,80,90,100,110,… 25:25,50,75,100,125,… 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. List the first several multiples of each number. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number. Look for the smallest number that is common to both lists. This number is the LCM. Find the LCM of 15 and 20 by listing multiples. List the first several multiples of 15 and of 20. Identify the first common multiple. 15:15,30,45,60,75,90,105,12020:20,40,60,80,100,120,140,160 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: 9and12 36 Find the least common multiple (LCM) of the given numbers: 18and24 72

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. 12=2⋅2⋅318=2⋅3⋅3 Then we write each number as a product of primes, matching primes vertically when possible. 12=2⋅2⋅3 18=2⋅3⋅3Now 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. Find 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. Find the LCM of 15 and 18 using the prime factors method. 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. LCM=2⋅3⋅3⋅5 The LCM of 15 and 18 is 90. Find the LCM using the prime factors method. 15and20 60 Find the LCM using the prime factors method. 15and35 105 Find the LCM of 50 and 100 using the prime factors method. 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. LCM=2⋅2⋅5⋅5 The LCM of 50 and 100 is 100. Find the LCM using the prime factors method: 55,88 440 Find the LCM using the prime factors method: 60,72 360 Ex 1: Prime Factorization Ex 2: Prime Factorization Ex 3: Prime Factorization Ex 1: Prime Factorization Using Stacked Division Ex 2: Prime Factorization Using Stacked Division The Least Common Multiple Example: Determining the Least Common Multiple Using a List of Multiples Example: Determining the Least Common Multiple Using Prime Factorization

Find the prime factorization of a composite number using the tree method. 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. Find the prime factorization of a composite number using the ladder method. Divide the number by the smallest prime. Continue dividing by that prime until it no longer divides evenly. Divide by the next prime until it no longer divides evenly. Continue until the quotient is a prime. 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. List the first several multiples of each number. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number. Look for the smallest number that is common to both lists. This number is the LCM. Find the LCM using the prime factors method. Find 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.

Find the Prime Factorization of a Composite Number In the following exercises, find the prime factorization of each number using the factor tree method. 86 2 ⋅ 43 78 132 2 ⋅ 2 ⋅ 3 ⋅ 11 455 693 3 ⋅ 3 ⋅ 7 ⋅ 11 420 115 5 ⋅ 23 225 2475 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 11 1560 In the following exercises, find the prime factorization of each number using the ladder method. 56 2 ⋅ 2 ⋅ 2 ⋅ 7 72 168 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7 252 391 17 ⋅ 23 400 432 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 627 2160 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 2520 In the following exercises, find the prime factorization of each number using any method. 150 2 ⋅ 3 ⋅ 5 ⋅ 5 180 525 3 ⋅ 5 ⋅ 5 ⋅ 7 444 36 2 ⋅ 2 ⋅ 3 ⋅ 3 50 350 2 ⋅ 5 ⋅ 5 ⋅ 7 144 Find the Least Common Multiple (LCM) of Two Numbers In the following exercises, find the least common multiple (LCM) by listing multiples. 8,12 24 4,3 6,15 30 12,16 30,40 120 20,30 60,75 300 44,55 In the following exercises, find the least common multiple (LCM) by using the prime factors method. 8,12 24 12,16 24,30 120 28,40 70,84 420 84,90 In the following exercises, find the least common multiple (LCM) using any method. 6,21 42 9,15 24,30 120 32,40

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!) 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?

ⓐ 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?

Use Variables and Algebraic Symbols In the following exercises, translate from algebra to English. 3⋅8 3 times 8, the product of three and eight. 12−x 24÷6 24 divided by 6, the quotient of twenty-four and six. 9+2a 50≥47 50 is greater than or equal to 47 3y<15 n+4=13 The sum of n and 4 is equal to 13 32−k=7 Identify Expressions and Equations In the following exercises, determine if each is an expression or equation. 5+u=84 equation 36−6s 4y−11 expression 10x=120 Simplify Expressions with Exponents In the following exercises, write in exponential form. 2⋅2⋅2 2³ a⋅a⋅a⋅a⋅a x⋅x⋅x⋅x⋅x⋅x x⁶ 10⋅10⋅10 In the following exercises, write in expanded form. 84 8 ⋅ 8 ⋅ 8 ⋅ 8 36 y5 y ⋅ y ⋅ y ⋅ y ⋅ y n4 In the following exercises, simplify each expression. 34 81 106 27 128 43 Simplify Expressions Using the Order of Operations In the following exercises, simplify. 10+2⋅5 20 (10+2)⋅5 (30+6)÷2 18 30+6÷2 72+52 74 (7+5)2 4+3(10−1) 31 (4+3)(10−1)

Evaluate an Expression In the following exercises, evaluate the following expressions. 9x−5whenx=7 58 y3wheny=5 3a−4bwhena=10,b=1 26 bhwhenb=7,h=8 Identify Terms, Coefficients and Like Terms In the following exercises, identify the terms in each expression. 12n2+3n+1 12n²,3n, 1 4x3+11x+3 In the following exercises, identify the coefficient of each term. 6y 6 13x2 In the following exercises, identify the like terms. 5x2,3,5y2,3x,x,4 3 and 4; 3x and x 8,8r2,8r,3r,r2,3s Simplify Expressions by Combining Like Terms In the following exercises, simplify the following expressions by combining like terms. 15a+9a 24a 12y+3y+y 4x+7x+3x 14x 6+5c+3 8n+2+4n+9 12n + 11 19p+5+4p−1+3p 7y2+2y+11+3y2−8 10y² + 2y + 3 13x2−x+6+5x2+9x Translate English Phrases to Algebraic Expressions In the following exercises, translate the following phrases into algebraic expressions. the difference of x and 6 x − 6 the sum of 10 and twice a the product of 3n and 9 3n ⋅ 9 the quotient of s and 4 5 times the sum of y and 1 5(y + 1) 10 less than the product of 5 and z Jack bought a sandwich and a coffee. The cost of the sandwich was $3 more than the cost of the coffee. Call the cost of the coffee c. Write an expression for the cost of the sandwich. c + 3 The number of poetry books on Brianna’s bookshelf is 5 less than twice the number of novels. Call the number of novels n. Write an expression for the number of poetry books.

Determine Whether a Number is a Solution of an Equation In the following exercises, determine whether each number is a solution to the equation. y+16=40 ⓐ 24 ⓑ 56 ⓐ yesⓑ no d−6=21 ⓐ 15 ⓑ 27 4n+12=36 ⓐ 6 ⓑ 12 ⓐ yesⓑ no 20q−10=70 ⓐ 3 ⓑ 4 15x−5=10x+45 ⓐ 2 ⓑ 10 ⓐ noⓑ yes 22p−6=18p+86 ⓐ 4 ⓑ 23 Model the Subtraction Property of Equality In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality. x + 3 = 5; x = 2 Solve Equations using the Subtraction Property of Equality In the following exercises, solve each equation using the subtraction property of equality. c+8=14 c = 6 v+8=150 23=x+12 x = 11 376=n+265 Solve Equations using the Addition Property of Equality In the following exercises, solve each equation using the addition property of equality. y−7=16 y = 23 k−42=113 19=p−15 p = 34 501=u−399 Translate English Sentences to Algebraic Equations In the following exercises, translate each English sentence into an algebraic equation. The sum of 7 and 33 is equal to 40. 7 + 33 = 40 The difference of 15 and 3 is equal to 12. The product of 4 and 8 is equal to 32. 4 ⋅ 8 = 32 The quotient of 63 and 9 is equal to 7. Twice the difference of n and 3 gives 76. 2(n − 3) = 76 The sum of five times y and 4 is 89. Translate to an Equation and Solve In the following exercises, translate each English sentence into an algebraic equation and then solve it. Eight more than x is equal to 35. x + 8 = 35; x = 27 21 less than a is 11. The difference of q and 18 is 57. q − 18 = 57; q = 75 The sum of m and 125 is 240. Mixed Practice In the following exercises, solve each equation. h−15=27 h = 42 k−11=34 z+52=85 z = 33 x+93=114 27=q+19 q = 8 38=p+19 31=v−25 v = 56 38=u−16

Identify Multiples of Numbers In the following exercises, list all the multiples less than 50 for each of the following. 3 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48 2 8 8, 16, 24, 32, 40, 48 10 Use Common Divisibility TestsIn the following exercises, using the divisibility tests, determine whether each number is divisible by 2,by3,by5,by6,and by10. 96 2, 3, 6 250 420 2, 3, 5, 6, 10 625 Find All the Factors of a Number In the following exercises, find all the factors of each number. 30 1, 2, 3, 5, 6, 10, 15, 30 70 180 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 378 Identify Prime and Composite Numbers In the following exercises, identify each number as prime or composite. 19 prime 51 121 composite 219

Find the Prime Factorization of a Composite Number In the following exercises, find the prime factorization of each number. 84 2 ⋅ 2 ⋅ 3 ⋅ 7 165 350 2 ⋅ 5 ⋅ 5 ⋅ 7 572 Find the Least Common Multiple of Two Numbers In the following exercises, find the least common multiple of each pair of numbers. 9,15 45 12,20 25,35 175 18,40

Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month. Answers will vary

In the following exercises, translate from an algebraic equation to English phrases. 6⋅4 15−x 15 minus x, the difference of fifteen and x. In the following exercises, identify each as an expression or equation. 5⋅8+10 x+6=9 equation 3⋅11=33 ⓐ Write n⋅n⋅n⋅n⋅n⋅n in exponential form. ⓑ Write 35 in expanded form and then simplify. ⓐ n⁶ⓑ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243 In the following exercises, simplify, using the order of operations. 4+3⋅5 (8+1)⋅4 36 1+6(3−1) (8+4)÷3+1 5 (1+4)2 5[2+7(9−8)] 45 In the following exercises, evaluate each expression. 8x−3whenx=4 y3wheny=5 125 6a−2bwhena=5,b=7 hwwhenh=12,w=3 36 Simplify by combining like terms. ⓐ 6x+8x ⓑ 9m+10+m+3 In the following exercises, translate each phrase into an algebraic expression. 5 more than x x + 5 the quotient of 12 and y three times the difference of aandb 3(a − b) Caroline has 3 fewer earrings on her left ear than on her right ear. Call the number of earrings on her right ear, r. Write an expression for the number of earrings on her left ear. In the following exercises, solve each equation. n−6=25 n = 31 x+58=71 In the following exercises, translate each English sentence into an algebraic equation and then solve it. 15 less than y is 32. y − 15 = 32; y = 47 the sum of a and 129 is 164. List all the multiples of 4 that are less than 50. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 Find all the factors of 90. Find the prime factorization of 1080. 2³ ⋅ 3³ ⋅ 5 Find the LCM (Least Common Multiple) of 24 and 40.

least common multiple The smallest number that is a multiple of two numbers is called the least common multiple (LCM). prime factorization The prime factorization of a number is the product of prime numbers that equals the number.