1.1 Whole Numbers

Izabela Mazur

CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra

1.1 Whole Numbers

Learning Objectives

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

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

As we begin our study of intermediate algebra, we need to refresh some of our skills and vocabulary. This chapter and the next 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.

$\phantom{\rule{1.5em}{0ex}}$ Counting Numbers: 1, 2, 3, …

$\phantom{\rule{1.5em}{0ex}}$ Whole Numbers: 0, 1, 2, 3, …

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

We can visualize counting numbers and whole numbers on a number line .See Figure 1.

The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left. While this number line shows only the whole numbers 0 through 6, the numbers keep going without end.

A horizontal number line with arrows on each end and values of zero to six runs along the bottom of the diagram. A second horizontal line with a left-facing arrow lies above the first and extend from zero to three. This line is labled “smaller”. A third horizontal line with a right-facing arrow lies above the first two, but runs from three to six and is labeled “larger”. — Figure 1

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 2 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.

The number 5,278,194 is shown in the chart. The digit 5 is in the millions place. The digit 2 is in the hundred-thousands place. The digit 7 is in the ten-thousands place. The digit 8 is in the thousands place. The digit 1 is in the hundreds place. The digit 9 is in the tens place. The digit 4 is in the ones place.

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 2

EXAMPLE 1

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

7
0
1
6
3

Solution

Place the number in the place value chart:

a) The 7 is in the thousands place.
b) The 0 is in the ten thousands place.
c) The 1 is in the tens place.
d) The 6 is in the ten-millions place.
e) The 3 is in the millions place.

TRY IT 1.1

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

a) 2 b) 1 c) 4 d) 7 e) 5

Show answer

a) ten millions b) tens c) hundred thousands d) millions e) ones

TRY IT 1.2

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

a) 9 b) 4 c) 2 d) 6 e) 7

Show answer

a) billions b) ten thousands c) tens d) hundred thousands e) 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 Figure 3). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

In this figure, the numbers 74, 218 and 369 are listed in a row, separated by commas. Each number has a curly bracket beneath it with the word “millions” written below the number 74, “thousands” written below the number 218, and “ones” written below the number 369. A left-facing arrow points at these three words, labeling them “periods”. One row down is the number “74”, a right-facing arrow and the words “Seventy-four million” followed by a comma. The next row below is the number “218”, a right-facing arrow and the words “two hundred eighteen thousand” followed by a comma. On the bottom row is the number “369”, a right-facing arrow and the words “three hundred sixty-nine”. — Figure 3

HOW TO: Name a Whole Number in Words.

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

EXAMPLE 2

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

Solution

Name the number in each period, followed by the period name.

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 2.1

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

Show answer

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

TRY IT 2.2

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

Show answer

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.

Identify the words that indicate periods. (Remember, the ones period is never named.)
Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
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

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. Separate the periods by commas.
Then write the digits in each period.

The number is 9,246,073,189.

TRY IT 3.1

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

Show answer

$2,466,714,051$

TRY IT 3.2

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

Show answer

$11,921,830,106$

In 2016, Statistics Canada estimated the population of Toronto as 13,448,494. We could say the population of Toronto was approximately 13.4 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 Toronto is approximately 13.4 million means that we rounded to the hundred thousands place.

EXAMPLE 4

Round 23,658 to the nearest hundred.

Solution

One row down, the instructions in the first cell say: “Step 2. Underline the digit to the right of the given place value.” In the second cell, the instructions say: “Underline the 5, which is to the right of the hundreds place.” In the third cell, there is the number 23,658 again, the same arrow pointing to the digit 6, labeling it the hundreds place. The 5 is also underlined in this cell. In the bottom row, the first cell says: “Step 4. Replace all digits to the right of the given place value with zeros. So, 23,700 is rounded to the nearest hundred.” In the second cell, the instructions say: “Replace all digits to the right of the hundreds place with zeros.” The third cell contains the number 23,700, which we have reached by rounding the number 23,658 to the nearest hundred.

TRY IT 4.1

Round to the nearest hundred: $17,852$ .

Show answer

$17,900$

TRY IT 4.2

Round to the nearest hundred: $468,751$ .

Show answer

$468,800$

HOW TO: Round Whole Numbers.

Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
Underline the digit to the right of the given place value.
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.
Replace all digits to the right of the given place value with zeros.

EXAMPLE 5

Round $103,978$ to the nearest:

hundred
thousand
ten thousand

Solution

a)

Locate the hundreds place in 103,978.
Underline the digit to the right of the hundreds place.
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.
	So, 104,000 is 103,978 rounded to the nearest hundred.

b)

Locate the thousands place and underline the digit to the right of the thousands place.
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.
	So, 104,000 is 103,978 rounded to the nearest thousand.

c)

Locate the ten thousands place and underline the digit to the right of the ten thousands place.	$\phantom{\rule{1.5em}{0ex}}$
Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros.	$\phantom{\rule{1.5em}{0ex}}$
	$\phantom{\rule{1.5em}{0ex}}$ So, 100,000 is 103,978 rounded to the nearest ten thousand.

TRY IT 5.1

Round 206,981 to the nearest: a) hundred b) thousand c) ten thousand.

Show answer

a) 207,000 b) 207,000 c) 210,000

TRY IT 5.2

Round 784,951 to the nearest: a) hundred b) thousand c) ten thousand.

Show answer

a) 785,000 b) 785,000 c) 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

A diagram made up of two rows of numbers. The top row reads “2, 4, 6, 8, 10, 12,” followed by an elipsis. Below 2 is 2 times 1, below 4 is 2 times 2, below 6 is 2 times 3, below 8 is 2 times 4, below 10 is 2 times 5, and below 12 is 2 times 6.

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

A diagram made up of two rows of numbers. The top row reads “3, 6, 9, 12, 15, 18,” followed by an elipsis. Below 3 is 3 times 1, below 6 is 3 times 2, below 9 is 3 times 3, below 12 is 3 times 4, below 15 is 3 times 5, and below 18 is 3 times 6.

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

The Table 1 below shows the multiples of 2 through 9 for the first 12 counting numbers.

Table 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 n if it is the product of a counting number and n.

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, $15\div 3$ is 5, so 15 is $5\cdot 3$ .

Divisible by a Number

If a number m is a multiple of n, then m is divisible by n.

Look at the multiples of 5 in Table 1. They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1 that shows 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.
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.

EXAMPLE 6

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

Solution

Is 5,625 divisible by 2?
Does it end in 0,2,4,6, or 8?	No. 5,625 is not divisible by 2.
Is 5,625 divisible by 3?
What is the sum of the digits?	$5+6+2+5=18$
Is the sum divisible by 3?	Yes. 5,625 is divisble by 3.
Is 5,625 divisible by 5 or 10?
What is the last digit? It is 5.	5,625 is divisble by 5 but not by 10.
Is 5,625 divisible by 6?
Is it divisible by both 2 or 3?	No, 5,625 is not divisible by 2, so 5,625 is not divisible by 6.

TRY IT 6.1

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

Show answer

by 2, 3, and 6

TRY IT 6.2

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

Show answer

by 3 and 5

Find Prime Factorization and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. 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 $8\cdot 9=72$ , we say that 8 and 9 are factors of 72. When we write $72=8\cdot 9$ , we say we have factored 72

An image shows the equation 8 times 9 equals 72. Written below the expression 8 times 9 is a curly bracket and the word “factors” while written below 72 is a horizontal bracket and the word “product”.

Other ways to factor 72 are $1\cdot 72,2\cdot 36,3\cdot 24,4\cdot 18,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}6\cdot 12$ . Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72

Factors

If $a\cdot b=m$ , then a and b are factors of m.

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

Prime Number and Composite Number

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 Figure 4, with their factors. Make sure to agree with the “prime” or “composite” label for each!

A table is shown with eleven rows and seven columns. The first row is a header row, and each cell labels the contents of the column below it. In the header row, the first three cells read from left to right “Number”, “Factors”, and “Prime or Composite?” The entire fourth column is blank. The last three cells read from left to right “Number”, “Factor”, and “Prime or Composite?” again. In each subsequent row, the first cell contains a number, the second contains its factors, and the third indicates whether the number is prime or composite. The three columns to the left of the blank middle column contain this information for the number 2 through 10, and the three columns to the right of the blank middle column contain this information for the number 11 through 19. On the left side of the blank column, in the first row below the header row, the cells read from left to right: “2”, “1,2”, and “Prime”. In the next row, the cells read from left to right: “3”, “1,3”, and “Prime”. In the next row, the cells read from left to right: “4”, “1,2,4”, and “Composite”. In the next row, the cells read from left to right: “5”, “1,5”, and “Prime”. In the next row, the cells read from left to right: “6”, “1,2,3,6” and “Composite”. In the next row, the cells read from left to right: “7”, “1,7”, and “Prime”. In the next row, the cells read from left to right: “8”, “1,2,4,8”, and “Composite”. In the next row, the cells read from left to right: “9”, “1,3,9”, and “Composite”. In the bottom row, the cells read from left to right: “10”, “1,2,5,10”, and “Composite”. On the right side of the blank column, in the first row below the header row, the cells read from left to right: “11”, “1,11”, and “Prime”. In the next row, the cells read from left to right: “12”, “1,2,3,4,6,12”, and “Composite”. In the next row, the cells read from left to right: “13”, “1,13”, and “Prime”. In the next row, the cells read from left to right “14”, “1,2,7,14”, and “Composite”. In the next row, the cells read from left to right: “15”, “1,3,5,15”, and “Composite”. In the next row, the cells read from left to right: “16”, “1,2,4,8,16”, and “Composite”. In the next row, the cells read from left to right, “17”, “1,17”, and “Prime”. In the next row, the cells read from left to right, “18”, “1,2,3,6,9,18”, and “Composite”. In the bottom row, the cells read from left to right: “19”, “1,19”, and “Prime”. — Figure 4

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.

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

One row down, the instructions in the first cell say: “Step 2. If a factor is prime, that branch is complete. Circle the prime.” In the second cell, the instructions say: “2 is prime. Circle the prime.” In the third cell, the factor tree from step 1 is repeated, but the 2 at the bottom of the tree is now circled. In the bottom row, the first cell says: “Step 4. Write the composite number as the product of all the circled primes.” The second cell is left blank. The third cell contains the algebraic equation 48 equals 2 times 2 times 2 times 2 times 3.

We say $2\cdot 2\cdot 2\cdot 2\cdot 3$ 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 $6\cdot 8$ , the result would still be the same. Finish the prime factorization and verify this for yourself.

TRY IT 7.1

Find the prime factorization of 80.

Show answer

$2\cdot 2\cdot 2\cdot 2\cdot 5$

TRY IT 7.2

Find the prime factorization of 60.

Show answer

$2\cdot 2\cdot 3\cdot 5$

HOW TO: Find the Prime Factorization of a Composite Number.

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

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

$252=2\cdot 2\cdot 3\cdot 3\cdot 7$

TRY IT 8.1

Find the prime factorization of 126

Show answer

$2\cdot 3\cdot 3\cdot 7$

TRY IT 8.2

Find the prime factorization of 294

Show answer

$2\cdot 3\cdot 7\cdot 7$

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:

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 that 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.

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

EXAMPLE 9

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

Solution

Make lists of the first few multiples of 15 and of 20, and use them to find the least common multiple.
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 9.1

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

Show answer

36

TRY IT 9.2

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

Show answer

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

In the bottom row of the table, the first cell says: “Step 4: Multiply the factors.” The second cell is bank. The third cell contains the equation LCM equals 36.

Notice that the prime factors of 12 $\left(2\cdot 2\cdot 3\right)$ and the prime factors of 18 $\left(2\cdot 3\cdot 3\right)$ are included in the LCM $\left(2\cdot 2\cdot 3\cdot 3\right)$ . 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.

TRY IT 10.1

Find the LCM using the prime factors method: 9 and 12

Show answer

36

TRY IT 10.2

Find the LCM using the prime factors method: 18 and 24

Show answer

72

HOW TO: Find the Least Common Multiple Using the Prime Factors Method.

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

EXAMPLE 11

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

Solution

Find the primes of 24 and 36. Match primes vertically when possible.Bring down all columns.
Multiply the factors.
	The LCM of 24 and 36 is 72.

TRY IT 11.1

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

Show answer

84

TRY IT 11.2

Find the LCM using the prime factors method: 24 and 32

Show answer

96

Key Concepts

Place Value as in Figure 2.
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 $m$ is a multiple of $n$ , then $m$ is divisible by $n$ . (If 6 is a multiple of 3, then 6 is divisible by 3.)

factors: If $a·b=m$ , then $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are factors of $m$ . Since 3 · 4 = 12, 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 n if it is the product of a counting number and n.

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 labeled 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, ….

Practice Makes Perfect

Use Place Value with Whole Numbers

In the following exercises, find the place value of each digit in the given numbers.

1. 51,493 a) 1 b) 4 c) 9 d) 5 e) 3	2. 87,210 a) 2 b) 8 c) 0 d) 7 e) 1
3. 164,285 a) 5 b) 6 c) 1 d) 8 e) 2	4. 395,076 a) 5 b) 3 c) 7 d) 0 e) 9
5. 93,285,170 a) 9 b) 8 c) 7 d) 5 e) 3	6. 36,084,215 a) 8 b) 6 c) 5 d) 4 e) 3
7. 7,284,915,860,132 a) 7 b) 4 c) 5 d) 3 e) 0	8. 2,850,361,159,433 a) 9 b) 8 c) 6 d) 4 e) 2

In the following exercises, name each number using words.

9. 1,078	10. 5,902
11. 364,510	12. 146,023
13. 5,846,103	14. 1,458,398
15. 37,889,005	16. 62,008,465

In the following exercises, 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

In the following, round to the indicated place value.

25. Round to the nearest ten.

a) 386 b) 2,931

26. Round to the nearest ten.

a) 792 b) 5,647

27. Round to the nearest hundred.

a) 13,748 b) 391,794

28. Round to the nearest hundred.

a) 28,166 b) 481,628

29. Round to the nearest ten.

a) 1,492 b) 1,497

30. Round to the nearest ten.

a) 2,791 b) 2,795

31. Round to the nearest hundred.

a) 63,994 b) 63,940

32. Round to the nearest hundred.

a) 49,584 b) 49,548

In the following exercises, round each number to the nearest a) hundred, b) thousand, c) ten thousand.

33. 392,546	34. 619,348
35. 2,586,991	36. 4,287,965

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 5, 6, and 10

37. 84	38. 9,696
39. 75	40. 78
41. 900	42. 800
43. 986	44. 942
45. 350	46. 550
47. 22,335	48. 39,075

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

49. 86	50. 78
51. 132	52. 455
53. 693	54. 400
55. 432	56. 627
57. 2,160	58. 2,520

In the following exercises, find the least common multiple of the each pair of numbers using the multiples method.

59. 8, 12	60. 4, 3
61. 12, 16	62. 30, 40
63. 20, 30	64. 44, 55

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

65. 8, 12	66. 12, 16
67. 28, 40	68. 84, 90
69. 55, 88	70. 60, 72

Everyday Math

71. Writing a Check Jorge bought a car for $24,493. He paid for the car with a check. Write the purchase price in words.	72. Writing a Check Marissa’s kitchen remodeling cost $18,549. She wrote a check to the contractor. Write the amount paid in words.
73. Buying a Car Jorge bought a car for $24,493. Round the price to the nearest a) ten b) hundred c) thousand; and d) ten-thousand.	74. Remodeling a Kitchen Marissa’s kitchen remodeling cost $18,549, Round the cost to the nearest a) ten b) hundred c) thousand and d) ten-thousand.
75. Population The population of China was 1,339,724,852 on November 1, 2010. Round the population to the nearest a) billion b) hundred-million; and c) million.	76. Astronomy The average distance between Earth and the sun is 149,597,888 kilometres. Round the distance to the nearest a) hundred-million b) ten-million; and c) million.
77. Grocery Shopping Hot dogs are sold in packages of 10, 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 12 and party cups come in packs of eight. What is the smallest number that makes the plates and cups come out even?

Writing Exercises

79. What is the difference between prime numbers and composite numbers?	80. Give an everyday example where it helps to round numbers.
81. Explain in your own words how to find the prime factorization of a composite number, using any method you prefer.

Answers

1. a) thousands b) hundreds c) tens d) ten thousands e) ones	3. a) ones b) ten thousands c) hundred thousands d) tens e) hundreds	5. a) ten millions b) ten thousands c) tens d) thousands e) millions
7. a) trillions b) billions c) millions d) tens e) thousands	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	17. 412
19. 35,975	21. 11,044,167	23. 3,226,512,017
25. a) 390 b) 2,930	27. a) 13,700 b) 391,800	29. a) 1,490 b) 1,500
31. a) 64,000 b) 63,900	33. a) 392,500 b) 393,000 c) 390,000	35. a) 2,587,000 b) 2,587,000 c) 2,590,000
37. divisible by 2, 3, and 6	39. divisible by 3 and 5	41. divisible by 2, 3, 5, 6, and 10
43. divisible by 2	45. divisible by 2, 5, and 10	47. divisible by 3 and 5
49. $2\cdot 43$	51. $2\cdot 2\cdot 3\cdot 11$	53. $3\cdot 3\cdot 7\cdot 11$
55. $2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3$	57. $2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3\cdot 5$	59. 24
61. 48	63. 60	65. 24
67. 420	69. 440	71. twenty-four thousand, four hundred ninety-three dollars
73. a) $24,490 b) $24,500 c) $24,000 d) $20,000	75. a) 1,000,000,000 b) 1,300,000,000 c) 1,340,000,000	77. 40
79. Answers may vary.	81. Answers may vary.

Attributions

This chapter has been adapted from “Introduction to Whole Numbers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

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Use Place Value with Whole Numbers

Identify Multiples and Apply Divisibility Tests

Find Prime Factorization and Least Common Multiples

Key Concepts

Glossary

Practice Makes Perfect

Use Place Value with Whole Numbers

Identify Multiples and Factors

Find Prime Factorizations and Least Common Multiples

Everyday Math

Writing Exercises

Answers

Attributions

License

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