2.4 Find Multiples and Factors

Learning objectives

Learning objectives

• Identify multiples of numbers

• Use common divisibility tests

• Find all the factors of a number

• Identify prime and composite numbers

Identify Multiples of Numbers

Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: $2,4,6,8,10,12.$ She has $12$ shoes in her closet.

The numbers $2,4,6,8,10,12$ are called multiples of $2.$ Multiples of $2$ can be written as the product of a counting number and $2.$ The first six multiples of $2$ are given below.

A multiple of a number is the product of the number and a counting number. So a multiple of $3$ would be the product of a counting number and $3.$ Below are the first six multiples of $3.$

We can find the multiples of any number by continuing this process. Table 2.8 shows the multiples of $2$ through $9$ for the first twelve counting numbers.

Table

2.8

Recognizing the patterns for multiples of $2,5,10,\mathrm{and}\,3$ will be helpful to you as you continue in this course.

Figure 2.6 shows the counting numbers from $1$ to $50.$ Multiples of $2$ are highlighted. Do you notice a pattern?

The last digit of each highlighted number in Figure 2.6 is either $0,2,4,6,\mathrm{or}\,8.$ This is true for the product of $2$ and any counting number. So, to tell if any number is a multiple of $2$ look at the last digit. If it is $0,2,4,6,\,\mathrm{or}\,8,$ then the number is a multiple of $2.$

Example 2.40

Solution

1. Recall the divisibility rule. A whole number is a multiple of $2$ exactly when its last digit is $0$, $2$, $4$, $6$, or $8$.
2. Test part ⓐ. The last digit of $489$ is $9$, which is not in that list, so $489$ is not a multiple of $2$.
3. Test part ⓑ. The last digit of $3,714$ is $4$, which is in the list, so $3,714$ is a multiple of $2$.
4. Verify with division. Dividing $3,714$ by $2$ gives the whole number $1,857$, confirming the second conclusion.

Takeaway: A whole number is a multiple of $2$ exactly when its last digit is $0$, $2$, $4$, $6$, or $8$; thus $489$ is not, but $3,714$ is.

Now let’s look at multiples of $5.$ Figure 2.7 highlights all of the multiples of $5$ between $1$ and $50.$ What do you notice about the multiples of $5?$

All multiples of $5$ end with either $5$ or $0.$ Just like we identify multiples of $2$ by looking at the last digit, we can identify multiples of $5$ by looking at the last digit.

Example 2.41

Solution

1. Recall the divisibility rule. A whole number is a multiple of $5$ exactly when its last digit is $0$ or $5$.
2. Test part ⓐ. The last digit of $579$ is $9$, which is neither $0$ nor $5$, so $579$ is not a multiple of $5$.
3. Test part ⓑ. The last digit of $880$ is $0$, which satisfies the rule, so $880$ is a multiple of $5$.
4. Verify with division. Dividing $880$ by $5$ gives the whole number $176$, confirming the second conclusion.

Takeaway: A whole number is a multiple of $5$ exactly when its last digit is $0$ or $5$; therefore $579$ is not a multiple of $5$, while $880$ is.

Figure 2.8 highlights the multiples of $10$ between $1$ and $50.$ All multiples of $10$ all end with a zero.

Example 2.42

Solution

1. Recall the divisibility rule. A whole number is a multiple of $10$ exactly when its last digit is $0$.
2. Test part ⓐ. The last digit of $425$ is $5$, not $0$, so $425$ is not a multiple of $10$.
3. Test part ⓑ. The last digit of $350$ is $0$, so $350$ is a multiple of $10$.
4. Verify with division. Dividing $350$ by $10$ gives the whole number $35$, confirming the second conclusion.

Takeaway: A whole number is a multiple of $10$ exactly when its last digit is $0$, so $425$ is not a multiple of $10$ but $350$ is.

Figure 2.9 highlights multiples of $3.$ The pattern for multiples of $3$ is not as obvious as the patterns for multiples of $2,5,\mathrm{and}\,10.$

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of $3$ is based on the sum of the digits. If the sum of the digits of a number is a multiple of $3,$ then the number itself is a multiple of $3.$ See Table 2.9.

Table

2.9

Consider the number $42.$ The digits are $4$ and $2,$ and their sum is $4+2=6.$ Since $6$ is a multiple of $3,$ we know that $42$ is also a multiple of $3.$

Example 2.43

Solution

1. Recall the divisibility rule. A whole number is a multiple of $3$ exactly when the sum of its digits is a multiple of $3$.
2. Test part ⓐ. The digit sum of $645$ is $6+4+5=15$, and $15$ is a multiple of $3$, so $645$ is a multiple of $3$, confirmed by $3\cdot215=645$.
3. Test part ⓑ. The digit sum of $10,519$ is $1+0+5+1+9=16$, and $16$ is not a multiple of $3$, so $10,519$ is not a multiple of $3$.
4. Check part ⓑ. Dividing $10,519$ by $3$ leaves a remainder of $1$, so the quotient is not a whole number.

Takeaway: A whole number is a multiple of $3$ exactly when its digit sum is divisible by $3$: $645$ has digit sum $15$ and is a multiple, while $10,519$ has digit sum $16$ and is not.

Look back at the charts where you highlighted the multiples of $2,$ of $5,$ and of $10.$ Notice that the multiples of $10$ are the numbers that are multiples of both $2$ and $5.$ That is because $10=2\cdot{}5.$ Likewise, since $6=2\cdot{}3,$ the multiples of $6$ are the numbers that are multiples of both $2$ and $3.$

Use Common Divisibility Tests

Another way to say that $375$ is a multiple of $5$ is to say that $375$ is divisible by $5.$ In fact, $375\mathrm{÷}5$ is $75,$ so $375$ is $5\cdot{}75.$ Notice in Example 2.43 that $10,519$ is not a multiple $3.$ When we divided $10,519$ by $3$ we did not get a counting number, so $10,519$ is not divisible by $3.$

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. Table 2.10 summarizes divisibility tests for some of the counting numbers between one and ten.

Table

2.10

Example 2.44

Solution

1. Apply the test for $2$. The last digit of $1,290$ is $0$, which is even, so $1,290$ is divisible by $2$.
2. Apply the test for $3$. The digit sum is $1+2+9+0=12$, and $12$ is a multiple of $3$, so $1,290$ is divisible by $3$.
3. Apply the test for $5$. The last digit is $0$, so $1,290$ is divisible by $5$.
4. Apply the test for $10$. A number ending in $0$ is divisible by $10$, so $1,290$ is divisible by $10$.
5. Verify the quotients. The divisions by $2$, $3$, $5$, and $10$ give $645$, $430$, $258$, and $129$, all whole numbers.

Takeaway: Applying the divisibility tests shows that $1,290$ is divisible by $2$, $3$, $5$, and $10$ because its last digit is $0$ and its digit sum is $12$.

Example 2.45

Solution

1. Apply the test for $2$. The last digit of $5,625$ is $5$, which is not even, so $5,625$ is not divisible by $2$.
2. Apply the test for $3$. The digit sum is $5+6+2+5=18$, and $18$ is a multiple of $3$, so $5,625$ is divisible by $3$.
3. Apply the test for $5$. The last digit is $5$, so $5,625$ is divisible by $5$.
4. Apply the test for $10$. The last digit is not $0$, so $5,625$ is not divisible by $10$.
5. Verify the quotients. The divisions by $3$ and $5$ give the whole numbers $1,875$ and $1,125$, while dividing by $2$ or $10$ leaves a remainder.

Takeaway: The number $5,625$ is divisible by $3$ and $5$ because its digit sum is $18$ and it ends in $5$, but it is not divisible by $2$ or $10$ because it does not end in $0$.

Find All the Factors of a Number

There are often several ways to talk about the same idea. So far, we’ve seen that if $m$ is a multiple of $n,$ we can say that $m$ is divisible by $n.$ We know that $72$ is the product of $8$ and $9,$ so we can say $72$ is a multiple of $8$ and $72$ is a multiple of $9.$ We can also say $72$ is divisible by $8$ and by $9.$ Another way to talk about this is to say that $8$ and $9$ are factors of $72.$ When we write $72=8\cdot{}9$ we can say that we have factored $72.$

In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.

For example, suppose a choreographer is planning a dance for a ballet recital. There are $24$ dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.

In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of $24.$ Table 2.13 summarizes the different ways that the choreographer can arrange the dancers.

Table

2.13

What patterns do you see in Table 2.13? Did you notice that the number of groups times the number of dancers per group is always $24?$ This makes sense, since there are always $24$ dancers.

You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of $24,$ which are:

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with $1.$ If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

Example 2.46

Solution

1. Plan the factor search. Test counting numbers in increasing order and keep each divisor together with its whole-number quotient.
2. Record the factor pairs. The divisors $1,2,3,4,6,8,$ and $9$ divide $72$ exactly, giving the pairs $1$ and $72$, $2$ and $36$, $3$ and $24$, $4$ and $18$, $6$ and $12$, $8$ and $9$, and $9$ and $8$.
3. Stop the search. After the pair $8$ and $9$, the quotient is smaller than the divisor, so any further pair would only repeat factors already listed.
4. List the factors in order. Collecting both members of each pair gives $1,2,3,4,6,8,9,12,18,24,36,$ and $72$.

Takeaway: The complete factor list of $72$ is $1,2,3,4,6,8,9,12,18,24,36,72$; testing divisors only until the quotient becomes smaller prevents listing each pair twice.

Identify Prime and Composite Numbers

Some numbers, like $72,$ have many factors. Other numbers, such as $7,$ have only two factors: $1$ and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number $1$ is neither prime nor composite. It has only one factor, itself.

Figure 2.10 lists the counting numbers from $2$ through $20$ along with their factors. The highlighted numbers are prime, since each has only two factors.

The prime numbers less than $20$ are $2,3,5,7,11,13,17,\mathrm{and}\,19.$ There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than $1$ and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.

Example 2.47

Solution

1. Recall the primality test. A number is prime if its only factors are $1$ and itself, so test the primes up to its square root.
2. Test part ⓐ. The number $83$ is not divisible by $2$, $3$, $5$, $7$, or $11$, the primes below its square root, so $83$ has no factor other than $1$ and itself.
3. Test part ⓑ. The number $77$ divides exactly by $7$, giving $77=7\cdot11$, so $77$ has a factor other than $1$ and itself.
4. Classify both numbers. Therefore $83$ is prime while $77$ is composite because it is the product of two smaller counting numbers.

Takeaway: Testing the primes up to the square root finds no factor of $83$, so $83$ is prime, while $77=7\cdot11$ is composite.

Section 2.4 Exercises

Section 2.4 Exercises

Identify Multiples of Numbers

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

Use Common Divisibility Tests

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

Find All the Factors of a Number

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

Identify Prime and Composite Numbers

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

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

1. Determine whether each number is a multiple of $2:$

1. ⓐ $\,678\,$

2. ⓑ $\,21,493$

2. Determine whether each number is a multiple of $2:$

1. ⓐ $\,979\,$

2. ⓑ $\,17,780$

3. Determine whether each number is a multiple of $5.$

1. ⓐ $\,675\,$

2. ⓑ $\,1,578$

1. Determine whether each number is a multiple of $5.$

1. ⓐ $\,421\,$

2. ⓑ $\,2,690$

2. Determine whether each number is a multiple of $10:$

1. ⓐ $\,179\,$

2. ⓑ $\,3,540$

3. Determine whether each number is a multiple of $10:$

1. ⓐ $\,110\,$

2. ⓑ $\,7,595$

1. Determine whether each number is a multiple of $3:$

1. ⓐ $\,954\,$

2. ⓑ $\,3,742$

2. Determine whether each number is a multiple of $3:$

1. ⓐ $\,643\,$

2. ⓑ $\,8,379$

3. Determine whether the given number is divisible by $2,3,5,\mathrm{and}\,10.$

$6240$

1. Determine whether the given number is divisible by $2,3,5,\mathrm{and}\,10.$

$7248$

2. Determine whether the given number is divisible $\mathrm{by}\,2,3,5,\mathrm{and}\,10.$

$4962$

3. Determine whether the given number is divisible $\mathrm{by}\,2,3,5,\mathrm{and}\,10.$

$3765$

1. Find all the factors of the given number:

$96$

2. Find all the factors of the given number:

$80$

3. Identify the number as prime or composite:

$91$

1. Identify the number as prime or composite:

$137$

2. $2$

3. $4$

1. $6$

2. $8$

3. $10$

1. $84$

2. $75$

3. $168$

1. $900$

2. $896$

3. $375$

1. $350$

2. $1430$

3. $22,335$

1. $36$

2. $60$

3. $144$

1. $588$

2. $43$

3. $39$

1. $71$

2. $481$

3. $209$

1. $667$