1.2: Introduction to Whole Numbers

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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 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.

We can visualize counting numbers and whole numbers on a number line (see Figure $\PageIndex{1}$).

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 $\PageIndex{1}$: 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.

Doing the Manipulative Mathematics activity “Number Line-Part 1” will help you develop a better understanding of the counting numbers and the whole numbers.

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

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 $\PageIndex{2}$: The number $5278194$ 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.

Exercise $\PageIndex{1}$

In the number $63407218$, find the place value of each digit:

$7$
$0$
$1$
$6$
$3$

Answer

Place the number in the place value chart:

$This figure is a table illustrating the number 63,407,218 within the place value system. The table is shown with a header row, labeled “Place Value”, divided into a second header row labeled “Trillions”, “Billions”, “Millions”, “Thousands” and “Ones”. Under the header “Trillions” are three labeled columns, written from bottom to top, that read “Hundred trillions”, “Ten trillions” and “Trillions”. Under the header “Billions” are three labeled columns, written from bottom to top, that read “Hundred billions”, “Ten billions” and “Billions”. Under the header “Millions” are three labeled columns, written from bottom to top, that read “Hundred millions”, “Ten millions” and “Millions”. Under the header “Thousands” are three labeled columns, written from bottom to top, that read “Hundred thousands”, “Ten thousands” and “Thousands”. Under the header “Ones” are three labeled columns, written from bottom to top, that read “Hundreds”, “Tens” and “Ones”. From left to right, below the columns labeled “Ten millions”, “Millions”, “Hundred thousands”, “Ten thousands”, “Thousands”, “Hundreds”, “Tens”, and “Ones”, are the following values: 6, 3, 4, 0, 7, 2, 1, 8. This means there are 6 ten millions, 3 millions, 4 hundred thousands, 0 ten thousands, 7 thousands, 2 hundreds, 1 ten, and 8 ones in the number sixty-three million, four hundred seven thousand, two hundred eighteen.$

The $7$ is in the thousands place.
The $0$ is in the ten thousands place.
The $1$ is in the tens place.
The $6$ is in the ten-millions place.
The $3$ is in the millions place.

Exercise $\PageIndex{2}$

For the number $27493615$, find the place value of each digit:

Answer

ten millions
tens
hundred thousands
millions
ones

Exercise $\PageIndex{3}$

For the number $519711641328$, find the place value of each digit:

Answer

billions
ten thousands
tens
hundred thousands
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 $\PageIndex{3}$). The number $74218369$ 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 $\PageIndex{3}$

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.

Exercise $\PageIndex{4}$

Name the number $8165432098710$ using words.

Answer

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

$In this figure, the numbers 8, 165, 432, 098 and 710 are listed in a row, separated by commas. Each number has a horizontal bracket beneath with the word “trillions” written below the number 8, “billions” written below the number 165, “millions” written below the number 432, “thousands” written below the number 098, and “ones” written below the number 710. One row down is the number 8, a right-facing arrow and the words “Eight trillion” followed by a comma. On the next row below is the number 165, a right-facing arrow and the words “One hundred sixty-five billion” followed by a comma. On the next row below is the number 432, a right-facing arrow and the words “Four hundred thirty-two million” followed by a comma. On the next row below is the number “098”, a right-facing arrow and the words “Ninety-eight thousand” followed by a comma. On the bottom row is the number 710, a right-facing arrow and the words “Seven hundred ten”.$

Put the commas in to separate the periods.

So, $8165432098710$ is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Exercise $\PageIndex{5}$

Name the number 9,258,137,904,0619,258,137,904,061 using words.

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

Exercise $\PageIndex{6}$

Name the number 17,864,325,619,00417,864,325,619,004 using words.

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.

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.

Exercise $\PageIndex{7}$

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

Answer: 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.; $An image has two lines of text. The upper lines read “nine billion”, followed by a comma, and “two hundred forty six million”, also followed by a comma. The words “billion” and “million” are underlined and each phrase has a curly bracket underneath. The lower lines read “seventy three thousand”, followed by a comma, and “one hundred eighty nine”. The word “thousand” is underlined and each phrase has a curly bracket underneath.$; The number is 9,246,073,189.

Exercise $\PageIndex{8}$

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

Answer: 2,466,714,051

Exercise $\PageIndex{9}$

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

Answer: 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.

Exercise $\PageIndex{10}$ How to Round Whole Numbers

Round 23,658 to the nearest hundred.

Answer: $This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains the numbers corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Locate the given place value with an arrow. All digits to the left do not change.” In the the second cell, the instructions say: “Locate the hundreds place in 23,658.” In the third cell, there is the number 23,658 with an arrow pointing to the digit 6, labeling it “hundreds place.”$ $One row down, the instructions in the first cell say: “Step 2. Underline the digit to the right of the given place value.” In the second cell, the instructions say: “Underline the 5, which is to the right of the hundreds place.” In the third cell, there is the number 23,658 again, the same arrow pointing to the digit 6, labeling it the hundreds place. The 5 is also underlined in this cell.$ $One row down, the first cell says: “Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value.” In the second cell, the instructions say: “Add 1 to the 6 in the hundreds place, since 5 is greater than or equal to 5.” The third cell contains the number 23,658 again, with an arrow pointing at the digit 6 and the text “add 1”. There is also a curly bracket under the digits 5 and 8, with an arrow pointing at them and the text “replace with 0s.”$ $In the bottom row, the first cell says: “Step 4. Replace all digits to the right of the given place value with zeros. So, 23,700 is rounded to the nearest hundred.” In the second cell, the instructions say: “Replace all digits to the right of the hundreds place with zeros.” The third cell contains the number 23,700, which we have reached by rounding the number 23,658 to the nearest hundred.$

Exercise $\PageIndex{11}$

Round to the nearest hundred: 17,852.

Answer: 17,900

Exercise $\PageIndex{12}$

Round to the nearest hundred: 468,751.

Answer: 468,800

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 11 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.

Exercise $\PageIndex{13}$

Round 103,978103,978 to the nearest:

hundred
thousand
ten thousand

Answer

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.

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.

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

Exercise $\PageIndex{14}$

Round 206,981 to the nearest: 1. hundred 2. thousand 3. ten thousand.

Answer

207,000
207,000
210,000

Exercise $\PageIndex{15}$

Round 784,951 to the nearest: 1. hundred 2. thousand 3. ten thousand.

Answer

785,000
785,000
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. — Figure $\PageIndex{4}$

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. — Figure $\PageIndex{5}$

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

Note

Doing the Manipulative Mathematics activity “Multiples” will help you develop a better understanding of multiples.

Table $\PageIndex{1}$ shows the multiples of 2 through 9 for the first 12 counting numbers.

Table $\PageIndex{1}$
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 $5\cdot3$.

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 $\PageIndex{1}$. 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 $\PageIndex{1}$ 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.

Exercise $\PageIndex{16}$

Is 5625 divisible by 2? By 3? By 5? By 6? By 10?

Answer

\[\begin{array} {ll} {\text{Is 5625 divisible by 2?}} &{} \\ {\text{Does it end in 0, 2, 4, 6, or 8?}} &{\text{No.}} \\ {} &{\text{5625 is not divisible by 2.}} \end{array}\]

\[\begin{array} {ll} {\text{Is 5625 divisible by 3?}} &{} \\ {\text{What is the sum of the digits?}} &{5 + 6 + 2 + 5 = 18} \\ {\text{Is the sum divisible by 3?}} &{\text{Yes, 5625 is divisible by 3.}} \end{array}\]

\[\begin{array} {ll} {\text{Is 5625 divisible by 5 or 10?}} &{} \\ {\text{What is the last digit? It is 5.}} &{\text{5625 is divisible by 5 but not by 10.}} \end{array}\]

\[\begin{array} {ll} {\text{Is 5625 divisible by 6?}} &{} \\ {\text{Is it divisible by both 2 and 3?}} &{\text{No, 5625 is not divisible by 2, so 5625 is }} \\ {} &{\text{not divisible by 6.}}\end{array}\]

Exercise $\PageIndex{17}$

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

Answer: by 2, 3, and 6

Exercise $\PageIndex{18}$

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

Answer: 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 $m$ 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.

Other ways to factor 72 are $1\cdot 72$, $2\cdot 36$, $3\cdot 24$, $4\cdot 18$ and $6\cdot 12$. 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.

Note

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.

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.

Note

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better understanding of prime numbers.

The counting numbers from 2 to 19 are listed in Figure $\PageIndex{7}$, 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 $\PageIndex{7}$

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.

Exercise $\PageIndex{19}$

Factor 48.

Answer

$This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions and some math. The third column contains most of the math work corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Find two factors whose product is the given number. Use these numbers to create two branches.” The second cell contains the algebraic equation 48 equals 2 times 24. In the third cell, there is a factor tree with 48 at the top. Two branches descend from 48 and terminate at 2 and 24 respectively.$ $One row down, the instructions in the first cell say: “Step 2. If a factor is prime, that branch is complete. Circle the prime.” In the second cell, the instructions say: “2 is prime. Circle the prime.” In the third cell, the factor tree from step 1 is repeated, but the 2 at the bottom of the tree is now circled.$ $One row down, the first cell says: “Step 3. If a factor is not prime, write it as the product of two factors and continue the process.” In the second cell, the instructions say: “24 is not prime. Break it into 2 more factors.” The third cell contains the original factor tree, with 48 at the top and two downward-pointing branches terminating at 2, which is underlined, and 24. Two more branches descend from 24 and terminate at 4 and 6 respectively. One line down, the instructions in the middle of the cell say “4 and 6 are not prime. Break them each into two factors.” In the cell on the right, the factor tree is repeated once more. Two branches descend from the 4 and terminate at 2 and 2. Both 2s are circled. Two more branches descend from 6 and terminate at a 2 and a 3, which are both circled. The instructions on the left say “2 and 3 are prime, so circle them.”$ $In the bottom row, the first cell says: “Step 4. Write the composite number as the product of all the circled primes.” The second cell is left blank. The third cell contains the algebraic equation 48 equals 2 times 2 times 2 times 2 times 3.$

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.

Exercise $\PageIndex{20}$

Find the prime factorization of 80.

Answer: $2\cdot 2\cdot 2\cdot 2\cdot 5$

Exercise $\PageIndex{21}$

Find the prime factorization of 60.

Answer: $2\cdot 2\cdot 3\cdot 5$

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.

Exercise $\PageIndex{22}$

Find the prime factorization of 252.

Answer

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

Exercise $\PageIndex{23}$

Find the prime factorization of 126.

Answer: $2\cdot 3\cdot 3\cdot 7$

Exercise $\PageIndex{24}$

Find the prime factorization of 294.

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:

Two rows of numbers are shown. The first row begins with 12, followed by a colon, then 12, 24, 36, 48, 60, 72, 84, 96, 108, and an elipsis. 36, 72, and 108 are bolded written in red. The second row begins with 18, followed by a colon, then 18, 36, 54, 72, 90, 108, and an elipsis. Again, the numbers 36, 72, and 108 are bolded written in red. On the line below is the phrase “Common Multiples”, a colon and the numbers 36, 72, and 108, written in red. One line below is the phrase “Least Common Multiple”, a colon and the number 36, written in blue. — Figure $\PageIndex{8}$

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.

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.

Exercise $\PageIndex{25}$

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

Answer

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.

Exercise $\PageIndex{26}$

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

Answer: $36$

Exercise $\PageIndex{27}$

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

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.

Exercise $\PageIndex{28}$

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

Answer: $This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions and some math. The third column contains most of the math work corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Write each number as a product of primes.” The second cell is left blank. In the third cell, there are two factor trees. In the first factor tree, two branches descend from 18 and terminate at 3 and 6 respectively. The 3 is prime and therefore circled. Two more branches descend from the 6 and terminate in 2 and 3, both of which are circled. In the second factor tree, two branches descend from 12 and terminate at 3 and 4. The 3 is circled. Two more branches descend from 4, terminating at 2 and 2, both of which are circled.$ $One row down, the instructions in the first cell say: “Step 2. List the primes of each number. Match primes vertically when possible.” In the second cell, the instructions say: “List the primes of 12. List the primes of 18. Line up with the primes of 12 when possible. If not create a new column.” The third cell contains the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12.$ $One row down, the instructions in the first cell say: “Bring down the number from each column.” The second cell is blank. The third cell contains the prime factorizations of 12 and 18 again, illustrated as two equations aligned just as they were before. This time, a horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18, ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18, ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18, ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.$ $In the bottom row of the table, the first cell says: “Step 4: Multiply the factors.” The second cell is bank. The third cell contains the equation LCM equals 36.$

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

Exercise $\PageIndex{29}$

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

Answer: $36$

Exercise $\PageIndex{30}$

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

Answer: $72$

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.

Exercise $\PageIndex{31}$

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

Answer

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.

Exercise $\PageIndex{32}$

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

Answer: $84$

Exercise $\PageIndex{33}$

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

Answer: $96$

Note

Access this online resource for additional instruction and practice with using whole numbers. You will need to enable Java in your web browser to use the application.

Sieve of Eratosthenes

Key Concepts

Place Value as in Figure.
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 $a\cdot b=m$, then $a$ and $b$ are factors of $m$. Since $3 \cdot 4 = 12$, 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, ....

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.	\(252=2\cdot 2\cdot 3\cdot 3\cdot 7\)

Learning Objectives

Use Place Value with Whole Numbers

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

NAME A WHOLE NUMBER IN WORDS.

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

WRITE A WHOLE NUMBER USING DIGITS.

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\) How to Round Whole Numbers

Exercise \(\PageIndex{11}\)

Exercise \(\PageIndex{12}\)

ROUND WHOLE NUMBERS.

Exercise \(\PageIndex{13}\)

Exercise \(\PageIndex{14}\)

Exercise \(\PageIndex{15}\)

Identify Multiples and Apply Divisibility Tests

Note

MULTIPLE OF A NUMBER

DIVISIBLE BY A NUMBER

DIVISIBILITY TESTS

Exercise \(\PageIndex{16}\)

Exercise \(\PageIndex{17}\)

Exercise \(\PageIndex{18}\)

Find Prime Factorizations and Least Common Multiples

FACTORS

Note

PRIME NUMBER AND COMPOSITE NUMBER

Note

PRIME FACTORIZATION

Exercise \(\PageIndex{19}\)

Exercise \(\PageIndex{20}\)

Exercise \(\PageIndex{21}\)

FIND THE PRIME FACTORIZATION OF A COMPOSITE NUMBER.

Exercise \(\PageIndex{22}\)

Exercise \(\PageIndex{23}\)

Exercise \(\PageIndex{24}\)

LEAST COMMON MULTIPLE

FIND THE LEAST COMMON MULTIPLE BY LISTING MULTIPLES.

Exercise \(\PageIndex{25}\)

Exercise \(\PageIndex{26}\)

Exercise \(\PageIndex{27}\)

Exercise \(\PageIndex{28}\)

Exercise \(\PageIndex{29}\)

Exercise \(\PageIndex{30}\)

FIND THE LEAST COMMON MULTIPLE USING THE PRIME FACTORS METHOD.

Exercise \(\PageIndex{31}\)

Exercise \(\PageIndex{32}\)

Exercise \(\PageIndex{33}\)

Note

Key Concepts

Glossary