# 2.4 Find multiples and factors  (Page 2/9)

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

1. $\phantom{\rule{0.2em}{0ex}}675\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}1,578$

1. yes
2. no

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

1. $\phantom{\rule{0.2em}{0ex}}421\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}2,690$

1. no
2. yes

[link] highlights the multiples of $10$ between $1$ and $50.$ All multiples of $10$ all end with a zero.

Determine whether each of the following is a multiple of $10\text{:}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}425\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}350$

## Solution

 ⓐ Is 425 a multiple of 10? Is the last digit zero? No. 425 is not a multiple of 10.
 ⓑ Is 350 a multiple of 10? Is the last digit zero? Yes. 350 is a multiple of 10.

Determine whether each number is a multiple of $10\text{:}$

1. $\phantom{\rule{0.2em}{0ex}}179\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}3,540$

1. no
2. yes

Determine whether each number is a multiple of $10\text{:}$

1. $\phantom{\rule{0.2em}{0ex}}110\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}7,595$

1. yes
2. no

[link] highlights multiples of $3.$ The pattern for multiples of $3$ is not as obvious as the patterns for multiples of $2,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.$

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 [link] .

 $\mathbf{\text{Multiple of 3}}$ $3$ $6$ $9$ $12$ $15$ $18$ $21$ $24$ $\mathbf{\text{Sum of digits}}$ $3$ $6$ $9$ $\begin{array}{c}\hfill 1+2\hfill \\ \hfill 3\hfill \end{array}$ $\begin{array}{c}\hfill 1+5\hfill \\ \hfill 6\hfill \end{array}$ $\begin{array}{c}\hfill 1+8\hfill \\ \hfill 9\hfill \end{array}$ $\begin{array}{c}\hfill 2+1\hfill \\ \hfill 3\hfill \end{array}$ $\begin{array}{c}\hfill 2+4\hfill \\ \hfill 6\hfill \end{array}$

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 $3.$

Determine whether each of the given numbers is a multiple of $3\text{:}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}645\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}10,519$

## Solution

Is $645$ a multiple of $3?$

 Find the sum of the digits. $6+4+5=15$ Is 15 a multiple of 3? Yes. If we're not sure, we could add its digits to find out. We can check it by dividing 645 by 3. $645÷3$ The quotient is 215. $3\cdot 215=645$

Is $10,519$ a multiple of $3?$

 Find the sum of the digits. $1+0+5+1+9=16$ Is 16 a multiple of 3? No. So 10,519 is not a multiple of 3 either.. $645÷3$ We can check this by dividing by 10,519 by 3. $\begin{array}{c}3,506\text{R}1\\ \hfill 3\overline{)10,519}\phantom{\rule{1em}{0ex}}\end{array}$

When we divide $10,519$ by $3,$ we do not get a counting number, so $10,519$ is not the product of a counting number and $3.$ It is not a multiple of $3.$

Determine whether each number is a multiple of $3\text{:}$

1. $\phantom{\rule{0.2em}{0ex}}954\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}3,742$

1. yes
2. no

Determine whether each number is a multiple of $3\text{:}$

1. $\phantom{\rule{0.2em}{0ex}}643\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}8,379$

1. no
2. yes

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, $375÷5$ is $75,$ so $375$ is $5\cdot 75.$ Notice in [link] 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 $3.$

## Divisibility

If a number $m$ is a multiple of $n,$ then we say that $m$ is divisible by $n.$

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

Divisibility Tests
A number is divisible by
$2$ if the last digit is $0,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}4,\phantom{\rule{0.2em}{0ex}}6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8$
$3$ if the sum of the digits is divisible by $3$
$5$ if the last digit is $5$ or $0$
$6$ if divisible by both $2$ and $3$
$10$ if the last digit is $0$

Determine whether $1,290$ is divisible by $2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.$

## Solution

[link] applies the divisibility tests to $1,290.$ In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.

Divisible by…? Test Divisible? Check
$2$ Is last digit $0,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}4,\phantom{\rule{0.2em}{0ex}}6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8?$ Yes. yes $1290÷2=645$
$3$ $\text{Is sum of digits divisible by}\phantom{\rule{0.2em}{0ex}}3?$
$1+2+9+0=12$ Yes.
yes $1290÷3=430$
$5$ Is last digit is $5$ or $0?$ Yes. yes $1290÷5=258$
$10$ Is last digit $0?$ Yes. yes $1290÷10=129$

Thus, $1,290$ is divisible by $2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.$

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