Using the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

1. $16 - 9$
   Show Solution

   16 minus 9, the difference of sixteen and nine
2. $25 - 7$
3. $5\cdot 6$
   Show Solution

   5 times 6, the product of five and six
4. $3\cdot 9$
5. $28\div 4$
   Show Solution

   28 divided by 4, the quotient of twenty-eight and four
6. $45\div 5$
7. $x+8$
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   x plus 8, the sum of x and eight
8. $x+11$
9. $\left(2\right)\left(7\right)$
   Show Solution

   2 times 7, the product of two and seven
10. $\left(4\right)\left(8\right)$
11. $14<21$
    Show Solution

    fourteen is less than twenty-one
12. $17<35$
13. $36\ge 19$
    Show Solution

    thirty-six is greater than or equal to nineteen
14. $42\ge 27$
15. $3n=24$
    Show Solution

    3 times n equals 24, the product of three and n equals twenty-four
16. $6n=36$
17. $y - 1>6$
    Show Solution

    y minus 1 is greater than 6, the difference of y and one is greater than six
18. $y - 4>8$
19. $2\le 18\div 6$
    Show Solution

    2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six
20. $3\le 20\div 4$
21. $a\ne 7\cdot 4$
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    a is not equal to 7 times 4, a is not equal to the product of seven and four
22. $a\ne 1\cdot 12$

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

1. $9\cdot 6=54$
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   equation
2. $7\cdot 9=63$
3. $5\cdot 4+3$
   Show Solution

   expression
4. $6\cdot 3+5$
5. $x+7$
   Show Solution

   expression
6. $x+9$
7. $y - 5=25$
   Show Solution

   equation
8. $y - 8=32$

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

1. $3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3$
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   3⁷
2. $4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4$
3. $x\cdot x\cdot x\cdot x\cdot x$
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   x⁵
4. $y\cdot y\cdot y\cdot y\cdot y\cdot y$

Simplify Expressions with Exponents

In the following exercises, write in expanded form.

1. ${5}^{3}$
   Show Solution

   125
2. ${8}^{3}$
3. ${2}^{8}$
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   256
4. ${10}^{5}$

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

1. $3+8\cdot 5$
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   43
2. $\text{(3+8)}\cdot \text{5}$
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   55
3. $2+6\cdot 3$
4. $\text{(2+6)}\cdot \text{3}$
5. ${2}^{3}-12\div \left(9 - 5\right)$
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   5
6. ${3}^{2}-18\div \left(11 - 5\right)$
7. $3\cdot 8+5\cdot 2$
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   34
8. $4\cdot 7+3\cdot 5$
9. $2+8\left(6+1\right)$
   Show Solution

   58
10. $4+6\left(3+6\right)$
11. $4\cdot 12/8$
    Show Solution

    6
12. $2\cdot 36/6$
13. $6+10/2+2$
    Show Solution

    13
14. $9+12/3+4$
15. $\left(6+10\right)\div \left(2+2\right)$
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    4
16. $\left(9+12\right)\div \left(3+4\right)$
17. $20\div 4+6\cdot 5$
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    35
18. $33\div 3+8\cdot 2$
19. $20\div \left(4+6\right)\cdot 5$
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    10
20. $33\div \left(3+8\right)\cdot 2$
21. ${4}^{2}+{5}^{2}$
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    41
22. ${3}^{2}+{7}^{2}$
23. ${\left(4+5\right)}^{2}$
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    81
24. ${\left(3+7\right)}^{2}$
25. $3\left(1+9\cdot 6\right)-{4}^{2}$
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    149
26. $5\left(2+8\cdot 4\right)-{7}^{2}$
27. $2\left[1+3\left(10 - 2\right)\right]$
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    50
28. $5\left[2+4\left(3 - 2\right)\right]$

Evaluating, Simplifying, and Translating Algebraic Expressions

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

1. $7x+8\text{ when }x=2$
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   22
2. $9x+7\text{ when }x=3$
3. $5x - 4\text{ when }x=6$
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   26
4.  $8x - 6\text{ when }x=7$
5. ${x}^{2}\text{ when }x=12$
   Show Solution

   144
6. ${x}^{3}\text{ when }x=5$
7. ${x}^{5}\text{ when }x=2$
   Show Solution

   32
8. ${x}^{4}\text{ when }x=3$
9. ${3}^{x}\text{ when }x=3$
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   27
10. ${4}^{x}\text{ when }x=2$
11. ${x}^{2}+3x - 7\text{ when }x=4$
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    21
12. ${x}^{2}+5x - 8\text{ when }x=6$
13. $2x+4y - 5\text{ when }x=7,y=8$
    Show Solution

    41
14. $6x+3y - 9\text{ when }x=6,y=9$
15. ${\left(x-y\right)}^{2}\text{ when }x=10,y=7$
    Show Solution

    9
16. ${\left(x+y\right)}^{2}\text{ when }x=6,y=9$
    Show Solution

    225
17. ${a}^{2}+{b}^{2}\text{ when }a=3,b=8$
    Show Solution

    73
18. ${r}^{2}-{s}^{2}\text{ when }r=12,s=5$
19. $2l+2w\text{ when }l=15,w=12$
    Show Solution

    54
20. $2l+2w\text{ when }l=18,w=14$

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

1. $15{x}^{2}+6x+2$
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   15×2, 6x, 2
2. $11{x}^{2}+8x+5$
3. $10{y}^{3}+y+2$
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   10y3, y, 2
4. $9{y}^{3}+y+5$

In the following exercises, identify the coefficient of the given term.

1. $8a$
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   8
2. $13m$
3. $5{r}^{2}$
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   5
4. $6{x}^{3}$

In the following exercises, identify all sets of like terms.

1. ${x}^{3},8x,14,8y,5,8{x}^{3}$
   Show Solution

   x3, 8×3 and 14, 5
2. $6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}$
3. $9a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2}$
   Show Solution

   16ab and 4ab; 16b2 and 9b2
4. $3,25{r}^{2},10s,10r,4{r}^{2},3s$

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

1. $10x+3x$
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   13x
2. $15x+4x$
3. $17a+9a$
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   26a
4. $18z+9z$
5. $4c+2c+c$
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   7c
6. $6y+4y+y$
7. $9x+3x+8$
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   12x + 8
8. $8a+5a+9$
9. $7u+2+3u+1$
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   10u + 3
10. $8d+6+2d+5$
11. $7p+6+5p+4$
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    12p + 10
12. $8x+7+4x - 5$
13. $10a+7+5a - 2+7a - 4$
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    22a + 1
14. $7c+4+6c - 3+9c - 1$
15. $3{x}^{2}+12x+11+14{x}^{2}+8x+5$
    Show Solution

    17×2 + 20x + 16
16. $5{b}^{2}+9b+10+2{b}^{2}+3b - 4$

Translate English Phrases into Algebraic Expressions

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

1. The sum of 8 and 12
   Show Solution

   8 + 12
2. The sum of 9 and 1
3. The difference of 14 and 9
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   14 − 9
4. 8 less than 19
5. The product of 9 and 7
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   9 ⋅ 7
6. The product of 8 and 7
7. The quotient of 36 and 9
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   36 ÷ 9
8. The quotient of 42 and 7
9. The difference of $x$ and $4$
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   x − 4
10. $3$ less than $x$
11. The product of $6$ and $y$
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    6y
12. The product of $9$ and $y$
13. The sum of $8x$ and $3x$
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    8x + 3x
14. The sum of $13x$ and $3x$
15. The quotient of $y$ and $3$
    Show Solution

    $\Large\frac{y}{3}$
16. The quotient of $y$ and $8$
17. Eight times the difference of $y$ and nine
    Show Solution

    8 (y − 9)
18. Seven times the difference of $y$ and one
19. Five times the sum of $x$ and $y$
    Show Solution

    5 (x + y)
20. Nine times five less than twice $x$

Translate English Phrases into Algebraic Expressions

In the following exercises, write an algebraic expression.

1. Adele bought a skirt and a blouse. The skirt cost $\$15$ more than the blouse. Let $b$ represent the cost of the blouse. Write an expression for the cost of the skirt.
   Show Solution

   b + 15
2. Eric has rock and classical CDs in his car. The number of rock CDs is $3$ more than the number of classical CDs. Let $c$ represent the number of classical CDs. Write an expression for the number of rock CDs.
3. The number of girls in a second-grade class is $4$ less than the number of boys. Let $b$ represent the number of boys. Write an expression for the number of girls.
   Show Solution

   b − 4
4. Marcella has $6$ fewer male cousins than female cousins. Let $f$ represent the number of female cousins. Write an expression for the number of boy cousins.
5. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.
   Show Solution

   2n − 7
6. Jeannette has $\$5$ and $\$10$ bills in her wallet. The number of fives is three more than six times the number of tens. Let $t$ represent the number of tens. Write an expression for the number of fives.

Subtraction Property of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each given value is a solution to the equation.

Exercise 1

$x+13=21$

1. $x=8$
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   yes
2. $x=34$
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   no

Exercise 2

$y+18=25$

1. $y=7$
2. $y=43$

Exercise 3

$m - 4=13$

1. $m=9$
   Show Solution

   no
2. $m=17$
   Show Solution

   yes

Exercise 4

$n - 9=6$

1. $n=3$
2. $n=15$

Exercise 5

$3p+6=15$

1. $p=3$
   Show Solution

   yes
2. $p=7$
   Show Solution

   no

Exercise 6

$8q+4=20$

1. $q=2$
2. $q=3$

Exercise 7

$18d - 9=27$

1. $d=1$
   Show Solution

   no
2. $d=2$
   Show Solution

   yes

Exercise 8

$24f - 12=60$

1. $f=2$
2. $f=3$

Exercise 9

$8u - 4=4u+40$

1. $u=3$
   Show Solution

   no
2. $u=11$
   Show Solution

   yes

Exercise 10

$7v - 3=4v+36$

1. $v=3$
2. $v=11$

Exercise 11

$20h - 5=15h+35$

1. $h=6$
   Show Solution

   no
2. $h=8$
   Show Solution

   yes

Exercise 12

$18k - 3=12k+33$

1. $k=1$
2. $k=6$

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

1. $a+2=18$
   Show Solution

   a = 16
2. $b+5=13$
3. $p+18=23$
   Show Solution

   p = 5
4. $q+14=31$
5. $r+76=100$
   Show Solution

   r = 24
6. $s+62=95$
7. $16=x+9$
   Show Solution

   x = 7
8. $17=y+6$
9. $93=p+24$
   Show Solution

   p = 69
10. $116=q+79$
11. $465=d+398$
    Show Solution

    d = 67
12. $932=c+641$

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

1. $y - 3=19$
   Show Solution

   y = 22
2. $x - 4=12$
3. $u - 6=24$
   Show Solution

   u = 30
4. $v - 7=35$
5. $f - 55=123$
   Show Solution

   f = 178
6. $g - 39=117$
7. $19=n - 13$
   Show Solution

   n = 32
8. $18=m - 15$
9. $10=p - 38$
   Show Solution

   p = 48
10. $18=q - 72$
11. $268=y - 199$
    Show Solution

    y = 467
12. $204=z - 149$

Translate Word Phrase to Algebraic Equations

In the following exercises, translate the given sentence into an algebraic equation.

1. The sum of $8$ and $9$ is equal to $17$.
   Show Solution

   8 + 9 = 17
2. The sum of $7$ and $9$ is equal to $16$.
3. The difference of $23$ and $19$ is equal to $4$.
   Show Solution

   23 − 19 = 4
4. The difference of $29$ and $12$ is equal to $17$.
5. The product of $3$ and $9$ is equal to $27$.
   Show Solution

   3 ⋅ 9 = 27
6. The product of $6$ and $8$ is equal to $48$.
7. The quotient of $54$ and $6$ is equal to $9$.
   Show Solution

   54 ÷ 6 = 9
8. The quotient of $42$ and $7$ is equal to $6$.
9. Twice the difference of $n$ and $10$ gives $52$.
   Show Solution

   2(n − 10) = 52
10. Twice the difference of $m$ and $14$ gives $64$.
11. The sum of three times $y$ and $10$ is $100$.
    Show Solution

    3y + 10 = 100
12. The sum of eight times $x$ and $4$ is $68$.

Translate to an Equation and Solve

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

1. Five more than $p$ is equal to $21$.
   Show Solution

   p + 5 = 21; p = 16
2. Nine more than $q$ is equal to $40$.
3. The sum of $r$ and $18$ is $73$.
   Show Solution

   r + 18 = 73; r = 55
4. The sum of $s$ and $13$ is $68$.
5. The difference of $d$ and $30$ is equal to $52$.
   Show Solution

   d − 30 = 52; d = 82
6. The difference of $c$ and $25$ is equal to $75$.
7. $12$ less than $u$ is $89$.
   Show Solution

   u − 12 = 89; u = 101
8. $19$ less than $w$ is $56$.
9. $325$ less than $c$ gives $799$.
   Show Solution

   c − 325 = 799; c = 1124
10. $299$ less than $d$ gives $850$.

Finding Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than $50$ for the given number.

1. $2$
   Show Solution

   2, 4, 6, 8, 10 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48
2. $3$
3. $4$
   Show Solution

   4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48
4. $5$
5. $6$
   Show Solution

   6, 12, 18, 24, 30, 36, 42, 48
6. $7$
7. $8$
   Show Solution

   8, 16, 24, 32, 40, 48
8. $9$
9. $10$
   Show Solution

   10, 20, 30, 40
10. $12$

Use Common Divisibility Tests

In the following exercises, use the divisibility tests to determine whether each number is divisible by $2,3,4,5,6,\text{and}10$.

1. $84$
   Show Solution

   Divisible by 2, 3, 4, 6
2. $96$
3. $75$
   Show Solution

   Divisible by 3, 5
4. $78$
5. $168$
   Show Solution

   Divisible by 2, 3, 4, 6
6. $264$
7. $900$
   Show Solution

   Divisible by 2, 3, 4, 5, 6, 10
8. $800$
9. $896$
   Show Solution

   Divisible by 2, 4
10. $942$
11. $375$
    Show Solution

    Divisible by 3, 5
12. $750$
13. $350$
    Show Solution

    Divisible by 2, 5, 10
14. $550$
15. $1430$
    Show Solution

    Divisible by 2, 5, 10
16. $1080$
17. $22,335$
    Show Solution

    Divisible by 3, 5
18. $39,075$

Find All the Factors of a Number

In the following exercises, find all the factors of the given number.

1. $36$
   Show Solution

   1, 2, 3, 4, 6, 9, 12, 18, 36
2. $42$
3. $60$
   Show Solution

   1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
4. $48$
5. $144$
   Show Solution

   1, 2, 3, 4, 6, 8, 12, 18, 24, 36, 48, 72,144
6. $200$
7. $588$
   Show Solution

   1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588
8. $576$

Identify Prime and Composite Numbers

In the following exercises, determine if the given number is prime or composite.

1. $43$
   Show Solution

   prime
2. $67$
3. $39$
   Show Solution

   composite
4. $53$
5. $71$
   Show Solution

   prime
6. $119$
7. $481$
   Show Solution

   composite
8. $221$
9. $209$
   Show Solution

   composite
10. $359$
11. $667$
    Show Solution

    composite
12. $1771$

Everyday Math

Banking

Frank’s grandmother gave him $\$100$ at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $\$15$ to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after graduation	Total number of dollars Frank put in the account	Simplified Total
$0$	$100$	$100$
$1$	$100+15$	$115$
$2$	$100+15\cdot 2$	$130$
$3$	$100+15\cdot 3$
$4$	$100+15\cdot \left[\right]$
$5$	$100+\left[\right]$
$6$
$20$
$x$

Show Solution

Banking

In March, Gina opened a Christmas club savings account at her bank. She deposited $\$75$ to open the account. Every week, she added $\$20$ to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after opening the account	Total number of dollars Gina put in the account	Simplified Total
$0$	$75$	$75$
$1$	$75+20$	$95$
$2$	$75+20\cdot 2$	$115$
$3$	$75+20\cdot 3$
$4$	$75+20\cdot \left[\right]$
$5$	$75+\left[\right]$
$6$
$20$
$x$

Writing Exercises

If a number is divisible by $2$ and by $3$, why is it also divisible by $6?$

What is the difference between prime numbers and composite numbers?

Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

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

1. $86$
   Show Solution

   2 ⋅ 43
2. $78$
3. $132$
   Show Solution

   2 ⋅ 2 ⋅ 3 ⋅ 11
4. $455$
5. $693$
   Show Solution

   3 ⋅ 3 ⋅ 7 ⋅ 11
6. $420$
7. $115$
   Show Solution

   5 ⋅ 23
8. $225$
9. $2475$
   Show Solution

   3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 11
10. 1560

Find the Prime Factorization of a Composite Number

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

1. $56$
   Show Solution

   2 ⋅ 2 ⋅ 2 ⋅ 7
2. $72$
3. $168$
   Show Solution

   2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7
4. $252$
5. $391$
   Show Solution

   17 ⋅ 23
6. $400$
7. $432$
   Show Solution

   2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3
8. $627$
9. $2160$
   Show Solution

   2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5
10. $2520$

Find the Prime Factorization of a Composite Number

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

1. $150$
   Show Solution

   2 ⋅ 3 ⋅ 5 ⋅ 5
2. $180$
3. $525$
   Show Solution

   3 ⋅ 5 ⋅ 5 ⋅ 7
4. $444$
5. $36$
   Show Solution

   2 ⋅ 2 ⋅ 3 ⋅ 3
6. $50$
7. $350$
   Show Solution

   2 ⋅ 5 ⋅ 5 ⋅ 7
8. $144$

Find the Least Common Multiple (LCM) of Two Numbers

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

1. $8,12$
   Show Solution

   24
2. $4,3$
3. $6,15$
   Show Solution

   30
4. $12,16$
5. $30,40$
   Show Solution

   120
6. $20,30$
7. $60,75$
   Show Solution

   300
8. $44,55$

Find the Least Common Multiple (LCM) of Two Numbers

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

1. $8,12$
   Show Solution

   24
2. $12,16$
3. $24,30$
   Show Solution

   120
4. $28,40$
5. $70,84$
   Show Solution

   420
6. $84,90$

Find the Least Common Multiple (LCM) of Two Numbers

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

1. $6,21$
   Show Solution

   42
2. $9,15$
3. $24,30$
   Show Solution

   120
4. $32,40$

Everyday Math

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!)

Show Solution

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!)

Writing Exercises

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?

Chapter Review Exercises