1.1 Introduction to Whole Numbers

Learning Objectives

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

  • Use place value with whole numbers
  • Identify multiples and apply divisibility tests
  • Find prime factorizations and least common multiples

As we begin our study of elementary algebra, we need to refresh some of our skills and vocabulary. This chapter will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

Use Place Value with Whole Numbers

The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting numbers. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole numbers.

Counting Numbers: 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, …

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly.

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 1.1 shows the place values. The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

This figure is a table illustrating the number 5,278,194 within the place value system. The table is shown with a header row, labeled “Place Value”, divided into a second header row labeled “Trillions”, “Billions”, “Millions”, “Thousands” and “Ones”. Under the header “Trillions” are three labeled columns, written from bottom to top, that read “Hundred trillions”, “Ten trillions” and “Trillions”. Under the header “Billions” are three labeled columns, written from bottom to top, that read “Hundred billions”, “Ten billions” and “Billions”. Under the header “Millions” are three labeled columns, written from bottom to top, that read “Hundred millions”, “Ten millions” and “Millions”. Under the header “Thousands” are three labeled columns, written from bottom to top, that read “Hundred thousands”, “Ten thousands” and “Thousands”. Under the header “Ones” are three labeled columns, written from bottom to top, that read “Hundreds”, “Tens” and “Ones”. From left to right, below the columns labeled “Millions”, “Hundred thousands”, “Ten thousands”, “Thousands”, “Hundreds”, “Tens”, and “Ones”, are the following values: 5, 2, 7, 8, 1, 9, 4. This means there are 5 millions, 2 hundred thousands, 7 ten thousands, 8 thousands, 1 hundreds, 9 tens, and 4 ones in the number five million two hundred seventy-nine thousand one hundred ninety-four."
Figure 1.1

Example 1

In the number 63,407,218, find the place value of each digit:

  1. 7
  2. 0
  3. 1
  4. 6
  5. 3
Solution

Place the number in the place value chart:

This figure is a table illustrating the number 5,278,194 within the place value system. The table is shown with a header row, labeled “Place Value”, divided into a second header row labeled “Trillions”, “Billions”, “Millions”, “Thousands” and “Ones”. Under the header “Trillions” are three labeled columns, written from bottom to top, that read “Hundred trillions”, “Ten trillions” and “Trillions”. Under the header “Billions” are three labeled columns, written from bottom to top, that read “Hundred billions”, “Ten billions” and “Billions”. Under the header “Millions” are three labeled columns, written from bottom to top, that read “Hundred millions”, “Ten millions” and “Millions”. Under the header “Thousands” are three labeled columns, written from bottom to top, that read “Hundred thousands”, “Ten thousands” and “Thousands”. Under the header “Ones” are three labeled columns, written from bottom to top, that read “Hundreds”, “Tens” and “Ones”. From left to right, below the columns labeled “Millions”, “Hundred thousands”, “Ten thousands”, “Thousands”, “Hundreds”, “Tens”, and “Ones”, are the following values: 5, 2, 7, 8, 1, 9, 4. This means there are 5 millions, 2 hundred thousands, 7 ten thousands, 8 thousands, 1 hundreds, 9 tens, and 4 ones in the number five million two hundred seventy-nine thousand one hundred ninety-four
Figure 1.2
  1. The 7 is in the thousands place.
  2. The 0 is in the ten-thousands place.
  3. The 1 is in the tens place.
  4. The 6 is in the ten-millions place.
  5. The 3 is in the millions place.

Try It

1) For the number 27,493,615, find the place value of each digit:

  1. 2
  2. 1
  3. 4
  4. 7
  5. 5
Solution
  1. ten millions
  2. tens
  3. hundred thousands
  4. millions
  5. ones

2) For the number 519,711,641,328, find the place value of each digit:

  1. 9
  2. 4
  3. 2
  4. 6
  5. 7
Solution
  1. billions
  2. ten thousands
  3. tens
  4. hundred thousands
  5. hundred millions

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see below). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

[latex]\underbrace{74\;}_{millions},\underbrace{218\;}_{thousands},\;\underbrace{369\;}_{ones}\xleftarrow{periods}\\74\xrightarrow[{}]{}Seventy-four\;million,\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;218\xrightarrow[{}]{}two\;hundred\;eighteen\;thousand,\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;369\xrightarrow[{}]{}three\;hundred\;sixty\;nine\\[/latex]

How to

Name a Whole Number in Words

  1. Start at the left and name the number in each period, followed by the period name.
  2. Put commas in the number to separate the periods.
  3. Do not name the ones period

 

Example 2 

Name the number 8,165,432,098,710 using words.

Solution

Step 1: Name the number in each period, followed by the period name.

[latex]\underbrace{\;\;\;\;8\;\;\;\;}_{trillions}\;,\underbrace{\;\;\;\;\;\;\;\;165\;\;\;\;\;\;}_{billions}\;,\underbrace{\;\;\;\;\;\;\;\;\;432\;\;\;\;\;\;\;}_{millions},\;\underbrace{\;\;\;\;\;\;098\;\;\;\;\;\;}_{thousands},\;\underbrace{\;\;\;\;\;\;710\;\;\;\;\;}_{ones}\;\;\;\\8\xrightarrow[{}]{}Eight\;trillion,\\165\xrightarrow[{}]{}One\;hundred\;sixty-five\;billion,\\432\xrightarrow[{}]{}Four\;hundred\;thirty-two\;million,\;\\098\xrightarrow[{}]{}Ninety-eight\;thousand,\\710\xrightarrow[{}]{}seven\;hundred\;ten[/latex]

Step 2: Put the commas in to separate the periods.

So, 8,165,432,098,710 is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Try It

3) Name the number 9,258,137,904,061 using words.

Solution

Nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one.

4) Name the number 17,864,325,619,004 using words.

Solution

Seventeen trillion, eight hundred sixty-four billion, three hundred twenty-five million, six hundred nineteen thousand four.

We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers, separating the periods with commas.

 

How to

Write a Whole Number Using Digits

    1. Identify the words that indicate periods. (Remember, the ones period is never named.)
    2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
    3. Name the number in each period and place the digits in the correct place value position.

 

Example 3

Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine as a whole number using digits.

Solution

Step 1: Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period.

Step 2: Separate the periods by commas.

Then write the digits in each period.

[latex]\underbrace{\;\;\;\;\;billions\;\;\;\;\;}_{nine\;billion}\;,\underbrace{\;\;\;\;\;\;\;\;\;\;\;millions\;\;\;\;\;\;\;\;\;\;}_{two\;hundred\;forty-six\;million}\;,\underbrace{\;\;\;\;\;\;\;\;\;\;thousands\;\;\;\;\;\;\;\;\;}_{seventy-three\;thousand},\;\underbrace{\;\;\;\;\;\;\;\;\;\;\;\;ones\;\;\;\;\;\;\;\;\;\;}_{one\;hundred\;eighty-nine},\;\\\;\;\;\;\;\;\;\;\;\\\begin{array}{ccc}\;\;\;\;&&\end{array}9\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;246\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;073\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;189\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[/latex]

The number is 9,246,073,189.

Try It

5) Write the number two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one as a whole number using digits.

Solution

2,466,714,051

6) Write the number eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six as a whole number using digits.

Solution

11,921,830,106

In 2013, the U.S. Census Bureau estimated the population of the state of New York as 19,651,127. We could say the population of New York was approximately 20 million. In many cases, you don’t need the exact value; an approximate number is good enough.

The process of approximating a number is called rounding. Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of New York is approximately 20 million means that we rounded to the millions place.

Example 4

Round 23,658 to the nearest hundred.

Solution

Step 1: Locate the given place value with an arrow. All digits to the left do not change. 

Locate the hundreds place 23,658.

[latex]{\color{red}{\text{Hundredths}}}\text{ place.}\;23,{\color{red}{\underset\uparrow6}}{58}[/latex]

Step 2: Underline the digit to the right of the given place value. 

Underline the 5, which is to the right of the hundreds place.

[latex]{\color{red}{\text{Hundredths}}}\text{ place.}\;23,{\color{red}{6}}{\color{blue}{\underline5}}{8}[/latex]

Step 3: Is this digit greater than or equal to 5?

Yes – add 1 to the digit in the given place value.

Add 1 to the 6 in the hundred place, since 5 is greater than or equal to 5.

[latex]{\text{Add 1.}\;23,}{\color{red}{6}}{\color{black}{5}}{8}[/latex]

Step 4: Replace all digits to the right of the given place with zeros.

Replace all digits to the right of the hundreds place with zeros.

[latex]{\text{Replace with zeros.}\;23,}{\color{blue}{700}}[/latex]

Try It

7) Round to the nearest hundred: 17,852.

Solution

17,900

8) Round to the nearest hundred: 468,751.

Solution

468,800

How to

Round Whole Numbers.

  1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
  2. Underline the digit to the right of the given place value.
  3. Is this digit greater than or equal to 5?
    • Yes–add 1  to the digit in the given place value.
    • No–do not change the digit in the given place value.
  4. Replace all digits to the right of the given place value with zeros.

 

Example 5

Round 103,978 to the nearest:

  1. hundred
  2. thousand
  3. ten thousand
Solution

a.

Step 1: Locate the hundreds place in 103,978.

[latex]{\color{red}{\text{Hundreds place:}}}{\;103,\boxed{\color{red}{9}}}{78}[/latex]

Step 2: Underline the digit to the right of the hundreds place.

[latex]{\color{red}{\text{Hundreds place:}}}{\;103,\boxed{\color{red}{9}}}{\underline78}[/latex]

Step 3: Since 7 is greater than or equal to 5, add 1 to the 9. Replace all digits to the right of the hundreds place with zeros.

[latex]{\color{red}{\text{Hundreds place:}}}{\;103,\boxed{\underbrace{\color{red}{9}}{_{Add\;1\;\;\\9+1=10\\replace\;9\;with\;0\\and\;carry\;the\;1}}}\underbrace{\underline78}_{\;replace\;with\;0s\\}=\;104,000}[/latex]

So, 104,000 is 103,978 rounded to the nearest hundred.


b.

Step 1: Locate the thousands place and underline the digit to the right of the thousands place.

[latex]{\color{red}{\text{Thousands place:}}}{\;10\;\boxed{\color{red}{3}}}{,\underline978}[/latex]

Step 2: Since 9 is greater than or equal to 5, add 1 to the 3. Replace all digits to the right of the hundreds place with zeros.

[latex]{\color{red}{\text{Thousands place:}}}{\;10\;\boxed{\underbrace{\color{red}{3}}{_{Add\;1\;\;\\3+1=4\\replace\;3\;with\;4}}}\underbrace{\underline978}_{\;replace\;with\;0s\\}=\;104,000}[/latex]

So, 104,000 is 103,978 rounded to the nearest thousand.


c.

Step 1: Locate the ten thousands place and underline the digit to the right of the ten thousands place.

[latex]{\color{red}{\text{Ten thousands place:}}}{\;1\boxed{\color{red}{0}}}{3,978}[/latex]

Step 2: Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros.

[latex]100,000[/latex]

So, 100,000 is 103,978 rounded to the nearest ten thousand.

Try It

9) Round 206,981 to the nearest:

  1. hundred
  2. thousand
  3. ten thousand.
Solution
  1. 207,000
  2. 207,000
  3. 210,000

10) Round 784,951 to the nearest:

  1. hundred
  2. thousand
  3. ten thousand.
Solution
  1. 785,000
  2. 785,000
  3. 780,000

Identify Multiples and Apply Divisibility Tests

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

2 2 · 1
4 2 · 2
6 2 · 3
8 2 · 4
10 2 · 5
12 2 · 6

Similarly, a multiple of 3 would be the product of a counting number and 3.

3 3 · 1
6 3 · 2
9 3 · 3
12 3 · 4
15 3 · 5
18 3 · 6

We could find the multiples of any number by continuing this process.

Table. 1.1 shows the multiples of 2 through 9 for the first 12 counting numbers.

Table 1.1
Counting Number 1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108
Multiples of 10 10 20 30 40 50 60 70 80 90 100 110 120

Multiple of a Number

A number is a multiple of [latex]n[/latex] if it is the product of a counting number and [latex]n[/latex].

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, [latex]15\div3[/latex] is [latex]5[/latex], so [latex]15[/latex] is [latex]15\times3[/latex].

Divisible by a Number

If a number [latex]m[/latex] is a multiple of [latex]n[/latex], then [latex]m[/latex] is divisible by [latex]n[/latex].

Look at the multiples of 5 in Table 1.1. They all end in 5 or 0. Numbers with the last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1.1 that show multiples of the numbers 2 through 9, we can discover the following divisibility tests:

Divisibility Tests

A number is divisible by:

  • 2 if the last digit is 0, 2, 4, 6, or 8.
  • 3 if the sum of the digits is divisible by 3.
  • 4 if the last two digits are divisible by 4.
  • 5 if the last digit is 5 or 0.
  • 6 if it is divisible by both 2 and 3.
  • 8 if it is divisible by 2 and 4.
  • 9 if the sum of the digits is divisible by 9.
  • 10 if it ends with 0.

Example 6

Is 5,625 divisible by 2? By 3? By 5? By 6? By 10?

Solution

Step 1: Is 5,625 divisible by 2?

No.

Step 2: Does it end in 0,2,4,6, or 8?

5,625 is not divisible by 2.

Step 3: Is 5,625 divisible by 3?

Yes. 5,625 is divisible by 3.

Step 4: What is the sum of the digits?

[latex]5+6+2+5=18[/latex]

Step 5: Is the sum divisible by 3?

Yes, 18 is divisible by 3.

Step 6: Is 5,625 divisible by 5 or 10?

5,625 is divisible by 5 but not by 10.

Step 7: What is the last digit?

It is 5.

Step 8: Is 5,625 divisible by 6?

No, it is not.

Step 9: Is it divisible by both 2 and 3?

No, 5,625 is not divisible by 2, but it is divisible by 3.

Try It

11) Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10.

Solution

By 2, 3, and 6.

12) Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10.

Solution

By 3 and 5.

Find Prime Factorizations and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if [latex]m[/latex] is a multiple of [latex]n[/latex], we can say that [latex]m[/latex] is divisible by [latex]n[/latex]. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since [latex]8\times9=72[/latex], we say that 8 and 9 are factors of 72. When we write [latex]72=8\times9[/latex], we say we have factored 72.

[latex]\underbrace{8\times9}_{factors}=\underbrace{72}_{product}[/latex]

Other ways to factor 72 are 1·72, 2·36, 3·24, 4·18, and 6·12. Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.

Factors

If [latex]a\cdot b=m[/latex] then [latex]a[/latex] and [latex]b[/latex] are factors of [latex]m[/latex].

Some numbers, like 72, have many factors. Other numbers have only two factors.

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

The counting numbers from 2 to 19 are listed in the below table, with their factors. Make sure to agree with the “prime” or “composite” label for each!

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number. These prime numbers are called the prime factors.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

Example 7

Factor 48.

Solution

4 and 8 are not prime. Break them each into two factors.
2 and 3 are prime, so circle them.

Step 1: Find two factors whose product is the given number. Use these numbers to create two branches.

[latex]48=2\times24\;\;\;\;\;\;\;\;\;\;\overbrace{2\;\;\;\;\;\;\;\;\;24}^{48}[/latex]

Step 2: If a factor is prime, that branch is complete. Circle the prime.

2 is prime. Circle the prime.

[latex]\overbrace{\boxed2\;\;\;\;\;\;\;\;\;\;\;\;\;\;24}^{48}[/latex]

Step 3: If a factor is not prime, write it as the product of two factors and continue the process.

24 is not prime break it into 2 more factors.4 and 6 are not prime. Break them each into two factors.

[latex]\overbrace{\boxed2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\overbrace{4\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6}^{24}}^{48}[/latex]

2 and 3 are prime, so circle them.

[latex]\overbrace{\boxed2\;\;\;\;\;\;\;\;\overbrace{\overbrace{\boxed2\;\;\;\;\;\;\;\;\boxed2}^4\;\;\;\;\;\;\;\;\;\;\;\;\overbrace{\boxed2\;\;\;\;\;\;\;\;\;\boxed3}^6}^{24}}^{48}[/latex]

Step 4: Write the composite number as the product of all the circled primes.

[latex]48=2\times2\times2\times2\times3[/latex]


We say 2·2·2·2·3 is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!

If we first factored 48 in a different way, for example as 6·8, the result would still be the same. Finish the prime factorization and verify this for yourself.

Try It

13) Find the prime factorization of 80. 

Solution

2·2·2·2·5

14) Find the prime factorization of 60.

Solution

2·2·3·5

How to

Find the Prime Factorization of a Composite Number.

  1. Find two factors whose product is the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
  3. If a factor is not prime, write it as the product of two factors and continue the process.
  4. Write the composite number as the product of all the circled primes.

 

Example 8

Find the prime factorization of 252.

Solution

Step 1: Find two factors whose product is 252. 12 and 21 are not prime.

Break 12 and 21 into two more factors. Continue until all primes are factored.

[latex]252\left\{\begin{array}{l}12\left\{\begin{array}{l}6\left\{\begin{array}{l}3\\2\end{array}\right.\\2\end{array}\right.\\21\left\{\begin{array}{l}3\\7\end{array}\right.\end{array}\right.[/latex]

Step 2: Write 252 as the product of all the circled primes.

[latex]252=2·2·3·3·7[/latex]

Try It

15) Find the prime factorization of 126.

Solution

2·3·3·7

16) Find the prime factorization of 294.

Solution

2·3·7·7

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. Two methods are used most often to find the least common multiple and we will look at both of them.

The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

12: 12, 24, 36, 48, 60, 72, 84, 96, 108

18: 18, 36, 54, 72, 90, 108

Common Multiples: 36, 72, 108

Least Common Multiple: 36

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18.

We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number which is a multiple of both numbers.

The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18.

How to

Find the Least Common Multiple by Listing Multiples.

  1. List several multiples of each number.
  2. Look for the smallest number that appears on both lists.
  3. This number is the LCM.

 

Example 9

Find the least common multiple of 15 and 20 by listing multiples.

Solution

Step 1: Make lists of the first few multiples of 15 and of 20, and use them to find the least common multiple.

15: 15, 30, 45, 60, 75, 90, 105, 120
20: 20, 40, 60, 80, 100, 120, 140, 160

Step 2: Look for the smallest number that appears in both lists.

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


Notice that 120 is in both lists, too. It is a common multiple, but it is not the least common multiple.

Try It

17) Find the least common multiple by listing multiples: 9 and 12.

Solution

36

18) Find the least common multiple by listing multiples: 18 and 24.

Solution

72

Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.

Example 10

Find the Least Common Multiple (LCM) of 12 and 18 using the prime factors method.

Solution

Step 1: Write each number as a product of primes.

[latex]\overbrace{\boxed3\;\;\;\;\;\;\;\;\;\overbrace{\boxed2\;\;\;\;\;\;\boxed{\;3}}^6}^{18}[/latex]
[latex]\overbrace{\boxed3\;\;\;\;\;\;\;\;\;\overbrace{\boxed2\;\;\;\;\;\;\boxed{\;2}}^4}^{12}[/latex]

Step 2: List the primes of each number. Match primes vertically when possible.

List the primes of 12.

[latex]12=2\times2\times3[/latex]

List the primes of 18. Line up with the primes of 12 when possible. If not create a new column.

[latex]18=2\times3\times3[/latex]

Step 3: Bring down the number from each column.

[latex]\begin{eqnarray*}12&=&2\times2\times3\\18&=&2\times\hspace{0.94cm}3\times3\\LCM&=&2\times2\times3\times3\end{eqnarray*}[/latex]

Step 4: Multiply the factors.

[latex]LCM=36[/latex]

Notice that the prime factors of 12 (2·2·3) and the prime factors of 18 (2·3·3) are included in the LCM (2·2·3·3). So 36 is the least common multiple of 12 and 18.

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

How to 

Find the Least Common Multiple Using the Prime Factors Method.

  1. Write each number as a product of primes.
  2. List the primes of each number. Match primes vertically when possible.
  3. Bring down the columns.
  4. Multiply the factors.

 

Try It

19) Find the LCM using the prime factors method: 9 and 12.

Solution

36

20) Find the LCM using the prime factors method: 18 and 24.

Solution

72

Example 11

Find the Least Common Multiple (LCM) of 24 and 36 using the prime factors method.

Solution

Step 1: Find the primes of 24 and 36.

Match primes vertically when possible.

Step 2: Bring down all columns.

[latex]\begin{eqnarray*}24&=&2\times2\times2\times3\\36&=&2\times2\hspace{0.94cm}\times3\times3\end{eqnarray*}[/latex]

Step 3: Multiply the factors.

[latex]\begin{eqnarray*}LCM&=&2\times2\times2\times3\times3\\LCM&=&72\end{eqnarray*}[/latex]

The LCM of 24 and 36 is 72.

Try It

21) Find the LCM using the prime factors method: 21 and 28.

Solution

84

22) Find the LCM using the prime factors method: 24 and 32.

Solution

96

Key Concepts

  • Place Value
  • Name a Whole Number in Words
    1. Start at the left and name the number in each period, followed by the period name.
    2. Put commas in the number to separate the periods.
    3. Do not name the ones period.
  • Write a Whole Number Using Digits
    1. Identify the words that indicate periods. (Remember the ones period is never named.)
    2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.
    3. Name the number in each period and place the digits in the correct place value position.
  • Round Whole Numbers
    1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
    2. Underline the digit to the right of the given place value.
    3. Is this digit greater than or equal to 5?
      • Yes—add 1 to the digit in the given place value.
      • No—do not change the digit in the given place value.
    4. Replace all digits to the right of the given place value with zeros.
  • Divisibility Tests: A number is divisible by:
    • 2 if the last digit is 0, 2, 4, 6, or 8.
    • 3 if the sum of the digits is divisible by 3.
    • 5 if the last digit is 5 or 0.
    • 6 if it is divisible by both 2 and 3.
    • 10 if it ends with 0.
  • Find the Prime Factorization of a Composite Number
    1. Find two factors whose product is the given number, and use these numbers to create two branches.
    2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
    3. If a factor is not prime, write it as the product of two factors and continue the process.
    4. Write the composite number as the product of all the circled primes.
  • Find the Least Common Multiple by Listing Multiples
    1. List several multiples of each number.
    2. Look for the smallest number that appears on both lists.
    3. This number is the LCM.
  • Find the Least Common Multiple Using the Prime Factors Method
    1. Write each number as a product of primes.
    2. List the primes of each number. Match primes vertically when possible.
    3. Bring down the columns.
    4. Multiply the factors.

Glossary

composite number
A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.
counting numbers
The counting numbers are the numbers 1, 2, 3, …
divisible by a number
If a number [latex]m[/latex] is a multiple of [latex]n[/latex], then [latex]m[/latex] is divisible by [latex]n[/latex]. (If 6 is a multiple of 3, then 6 is divisible by 3.)
factors
If [latex]a\;\times\;b\;=\;m[/latex], then ([latex]a[/latex] and [latex]b[/latex]) are factors of [latex]m[/latex]. Since [latex]3 \times 4 = 12[/latex], then 3 and 4 are factors of 12.
least common multiple
The least common multiple of two numbers is the smallest number that is a multiple of both numbers.
multiple of a number
A number is a multiple of [latex]n[/latex] if it is the product of a counting number and [latex]n[/latex].
number line
A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left.
origin
The origin is the point labelled 0 on a number line.
prime factorization
The prime factorization of a number is the product of prime numbers that equals the number.
prime number
A prime number is a counting number greater than 1, whose only factors are 1 and itself.
whole numbers
The whole numbers are the numbers 0, 1, 2, 3, ….

Exercises: Place Value With Whole Numbers 

Instructions:  For questions 1-8, find the place value of each digit in the given numbers.

1)   [latex]51\text{,}493[/latex]

a) [latex]1[/latex]
b) [latex]4[/latex]
c) [latex]9[/latex]
d) [latex]5[/latex]
e) [latex]3[/latex]

2)  [latex]87\text{,}210[/latex]
a)  [latex]2[/latex]

b)  [latex]8[/latex]
c)  [latex]0[/latex]
d)  [latex]7[/latex]
e)  [latex]1[/latex]

3)  [latex]164\text{,}285[/latex]
a)  [latex]5[/latex]

b)  [latex]6[/latex]
c)  [latex]1[/latex]
d)  [latex]8[/latex]
e)  [latex]2[/latex]

4)  [latex]395\text{,}076[/latex]
a)  [latex]5[/latex]

b)  [latex]3[/latex]
c)  [latex]7[/latex]
d)  [latex]0[/latex]
e)  [latex]9[/latex]

5)  [latex]93\text{,}285\text{,}170[/latex]
a)  [latex]9[/latex]

b)  [latex]8[/latex]
c)  [latex]7[/latex]
d)  [latex]5[/latex]
e)  [latex]3[/latex]

6)  [latex]36\text{,}084\text{,}215[/latex]

a)  [latex]8[/latex]
b)  [latex]6[/latex]
c)  [latex]5[/latex]
d)  [latex]4[/latex]
e)  [latex]3[/latex]

7)  [latex]7\text{,}284\text{,}915\text{,}860\text{,}132[/latex]

a)  [latex]7[/latex]
b)  [latex]4[/latex]
c)  [latex]5[/latex]
d)  [latex]3[/latex]
e)  [latex]0[/latex]

8)  [latex]2\text{,}850\text{,}361\text{,}159\text{,}433[/latex]

a)  [latex]9[/latex]
b)  [latex]8[/latex]
c)  [latex]6[/latex]
d)  [latex]4[/latex]
e)  [latex]2[/latex]


Odd Answers

1a)  thousands
1b)  hundreds
1c)  tens
1d)  ten thousands
1e) ones

3a)  ones
3b)  ten thousands
3c)  hundred thousands
3d)  tens
3e)  hundreds

5a)  ten millions
5b)  ten thousands
5c)  tens
5d)  thousands
5e)  millions

7a)  trillions
7b)  billions
7c)  millions
7d)  tens
7e)  thousands

 

Exercises: Name Numbers Using Words

Instructions:  For questions 9-16, name each number using words.

9) [latex]1\text{,}078[/latex]

10)  [latex]5\text{,}902[/latex]

11)  [latex]364\text{,}510[/latex]

12)  [latex]146\text{,}023[/latex]

13)  [latex]5\text{,}846\text{,}103[/latex]

14)  [latex]1\text{,}458\text{,}398[/latex]

15)  [latex]37\text{,}889\text{,}005[/latex]

16)  [latex]62\text{,}008\text{,}465[/latex]


Odd Answers

9) one thousand, seventy-eight

11)  three hundred sixty-four thousand, five hundred ten

13)  five million, eight hundred forty-six thousand, one hundred three

15)  thirty-seven million, eight hundred eighty-nine thousand, five
 

Exercises: Whole Numbers Using Digits

Instructions:  For questions 17-24, write each number as a whole number using digits.

17)  four hundred twelve

18)  two hundred fifty-three

19)  thirty-five thousand, nine hundred seventy-five

20)  sixty-one thousand, four hundred fifteen

21)  eleven million, forty-four thousand, one hundred sixty-seven

22)  eighteen million, one hundred two thousand, seven hundred eighty-three

23)  three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

24)  eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six


Odd Answers

17)  [latex]412[/latex]

19)  [latex]35\text{,}975[/latex]

21)  [latex]11\text{,}044\text{,}167[/latex]

23) [latex]3\text{,}226\text{,}512\text{,}017[/latex]

Exercises: Round to Indicated Place Value

Instructions:  For questions 25-32, round to the indicated place value.

25)  Round to the nearest ten.

a)  [latex]386[/latex]
b)  [latex]2\text{,}931[/latex]

26)  Round to the nearest ten.

a)  [latex]792[/latex]
b)  [latex]5\text{,}647[/latex]

27)  Round to the nearest hundred.

a)  [latex]13\text{,}748[/latex]
b)  [latex]391\text{,}794[/latex]

28)  Round to the nearest hundred.

a)  [latex]28\text{,}166[/latex]
b)  [latex]481\text{,}628[/latex]

29)  Round to the nearest ten.

a)  [latex]1\text{,}492[/latex]
b)  [latex]1\text{,}497[/latex]

30)  Round to the nearest ten.

a)  [latex]2\text{,}791[/latex]
b)  [latex]2\text{,}795[/latex]

31)  Round to the nearest hundred.

a)  [latex]63\text{,}994[/latex]
b)  [latex]63\text{,}940[/latex]

32)  Round to the nearest hundred.

a)  [latex]49\text{,}584[/latex]
b)  [latex]49\text{,}548[/latex]


Odd Answers

25a)  [latex]390[/latex]
25b)  [latex]2\text{,}930[/latex]

27a)  [latex]13\text{,}700[/latex]
27b)  [latex]391\text{,}800[/latex]

29a)  [latex]1\text{,}490[/latex]
29b) 
[latex]1\text{,}500[/latex]

31a)  [latex]64\text{,}000[/latex]
31b) 
[latex]63\text{,}900[/latex]

Exercises: Round Numbers to Nearest Hundred, Thousand, and Ten Thousand

Instructions:  For questions 33-36, round each number to the nearest:

a) hundred
b) thousand
c) ten thousand

33)  [latex]392\text{,}546[/latex]

34)  [latex]619\text{,}348[/latex]

35)  [latex]2\text{,}586\text{,}991[/latex]

36)  [latex]4\text{,}287\text{,}965[/latex]


Odd Answers

33a)  [latex]392\text{,}500[/latex]
33b)  [latex]393\text{,}000[/latex]
33c)  [latex]390\text{,}000[/latex]

35a)  [latex]2\text{,}587\text{,}000[/latex]
35b)  [latex]2\text{,}587\text{,}000[/latex]
35c)  [latex]2\text{,}590\text{,}000[/latex]

Exercises: Identify Multiples and Factors

Instructions:  For questions 37-48, use the divisibility tests to determine whether each number is divisible by [latex]2[/latex], [latex]3[/latex], [latex]5[/latex], [latex]6[/latex], and [latex]10[/latex].

37)  [latex]84[/latex]

38)  [latex]9\text{,}696[/latex]

39)  [latex]75[/latex]

40)  [latex]78[/latex]

41)  [latex]900[/latex]

42)  [latex]800[/latex]

43)  [latex]986[/latex]

44)  [latex]942[/latex]

45)  [latex]350[/latex]

46)  [latex]550[/latex]

47)  [latex]22\text{,}335[/latex]

48)  [latex]39\text{,}075[/latex]


Odd Answers

37)  divisible by [latex]2[/latex], [latex]3[/latex], and [latex]6[/latex]

39)  divisible by [latex]3[/latex] and [latex]5[/latex]

41)  divisible by [latex]2[/latex], [latex]3[/latex], [latex]5[/latex], [latex]6[/latex], and [latex]10[/latex]

43)  divisible by [latex]2[/latex]

45)  divisible by [latex]2[/latex], [latex]5[/latex], and [latex]10[/latex]

47)  divisible by [latex]3[/latex] and [latex]5[/latex]

Exercises: Find Prime Factorizations

 

 

 

Instructions:  For questions 49-59, find the prime factorization.

49)  [latex]86[/latex]

50)  [latex]78[/latex]

51)  [latex]132[/latex]

52)  [latex]243[/latex]

53)  [latex]693[/latex]

54)  [latex]455[/latex]

55)  [latex]432[/latex]

56)  [latex]400[/latex]

57)  [latex]2\text{,}160[/latex]

58)  [latex]627[/latex]

59)  [latex]2\text{,}560[/latex]


Odd Answers

49)  [latex]2,43[/latex] 

51)  [latex]2,2,3,11[/latex]

53)  [latex]3,3,7,11[/latex]

55)  [latex]2,2,2,2,3,3,3[/latex]

57)  [latex]2,2,2,2,3,3,3,5[/latex]

59) [latex]2,2,2,2,2,2,2,2,2,5[/latex]

Exercises: Find Least Common Multiples Using Multiples Method

 

Instructions:  For questions 60-64, find the least common multiple of each pair of numbers using the multiples method.

60)  [latex]44,55[/latex][latex]8,12[/latex]

61)  [latex]4,3[/latex]

62)  [latex]12,16[/latex]

63)  [latex]30,40[/latex]

64)  [latex]20,30[/latex]


Odd Answers

61)  [latex]12[/latex]

63)  [latex]120[/latex]

 

Exercises: Find Least Common Multiples using Prime Factors Method

 

Instructions:  For questions 65-70, find the least common multiple of each pair of numbers using the prime factors method.

65)  [latex]8,12[/latex]

66)  [latex]12,16[/latex]

67)  [latex]28,40[/latex]

68)  [latex]84,90[/latex]

69)  [latex]55,88[/latex]

70)  [latex]60,72[/latex]


Odd Answers

65)  [latex]24[/latex]

67)  [latex]280[/latex]

69)  [latex]440[/latex]

 

Exercises: Everyday Math

 

Instructions:  For questions 71-78, answer the everyday math word problems.

71)  Writing a Check Jorge bought a car for [latex]$24\text{,}493[/latex]. He paid for the car with a check. Write the purchase price in words.

72)  Writing a Check. Marissa’s kitchen remodeling cost [latex]$18\text{,}549[/latex]. She wrote a check to the contractor. Write the amount paid in words.

73)  Buying a Car: Jorge bought a car for [latex]$24\text{,}493[/latex]. Round the price to the nearest

a)  ten
b)  hundred
c)  thousand
d)  ten-thousand

74)  Remodeling a Kitchen.Marissa’s kitchen remodeling cost [latex]$18\text{,}549[/latex]. Round the cost to the nearest

a)  ten
b)  hundred
c)  thousand
d)  ten-thousand

75)  Population.The population of China was [latex]1\text{,}339\text{,}724\text{,}852[/latex] on November 1, 2010. Round the population to the nearest

a)  billion
b)  hundred-million
c)  million

76)  Astronomy.The average distance between Earth and the sun is [latex]149\text{,}597\text{,}888[/latex] kilometers. Round the distance to the nearest

a)  hundred-million
b) ten-million
c) million

77)  Grocery Shopping.Hot dogs are sold in packages of [latex]10[/latex], but hot dog buns come in packs of eight. What is the smallest number that makes the
hot dogs and buns come out even?

 

78)  Grocery Shopping.Paper plates are sold in packages of [latex]12[/latex] and party cups come in packs of eight. What is the smallest number that makes the plates and cups come out even?

Odd Answers

71)  twenty-four thousand, four hundred ninety-three dollars

73a)  [latex]$24\text{,}490[/latex]
73b)  [latex]$24\text{,}500[/latex]
73c)  [latex]$24\text{,}000[/latex]
73d)  [latex]$20\text{,}000[/latex]

75a)  [latex]1\text{,}000\text{,}000\text{,}000[/latex]
75b)  [latex]1\text{,}300\text{,}000\text{,}000[/latex]
75c)  [latex]1\text{,}340\text{,}000\text{,}000[/latex]

77)  [latex]40[/latex]

 

Exercises: Writing Exercises

Instructions:  For questions 79-82, answer the given writing exercises.

79)  Give an everyday example where it helps to round numbers.

80)  If a number is divisible by [latex]2[/latex] and by [latex]3[/latex] why is it also divisible by [latex]6[/latex]?

81)  What is the difference between prime numbers and composite numbers?

82)  Explain in your own words how to find the prime factorization of a composite number, using any method you prefer.


Odd Answers

79)  Answers may vary.

81)  Answers may vary.

 

 

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