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Determine whether each number is a multiple of $5.$
Determine whether each number is a multiple of $5.$
[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 $10\text{:}\phantom{\rule{0.2em}{0ex}}$
ⓐ | |
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 $10\text{:}$
Determine whether each number is a multiple of $10\text{:}$
[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 $3\text{:}\phantom{\rule{0.2em}{0ex}}$
ⓐ 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. | $645\xf73$ |
The quotient is 215. | $3\cdot 215=645$ |
ⓑ Is $\mathrm{10,519}$ 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.. | $645\xf73$ |
We can check this by dividing by 10,519 by 3. | $\begin{array}{c}\mathrm{3,506}\text{R}1\\ \hfill 3\overline{)\mathrm{10,519}}\phantom{\rule{1em}{0ex}}\end{array}$ |
When we divide $\mathrm{10,519}$ by $3,$ we do not get a counting number, so $\mathrm{10,519}$ 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 $3\text{:}$
Determine whether each number is a multiple of $3\text{:}$
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 $3.$
Another way to say that $375$ is a multiple of $5$ is to say that $375$ is divisible by $5.$ In fact, $375\xf75$ is $75,$ so $375$ is $5\cdot 75.$ Notice in [link] that $\mathrm{10,519}$ is not a multiple $3.$ When we divided $\mathrm{10,519}$ by $3$ we did not get a counting number, so $\mathrm{10,519}$ is not divisible by $3.$
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 $\mathrm{1,290}$ is divisible by $2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.$
[link] applies the divisibility tests to $\mathrm{1,290}.$ 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 | $1290\xf72=645$ |
$3$ |
$\text{Is sum of digits divisible by}\phantom{\rule{0.2em}{0ex}}3?$
$1+2+9+0=12$ Yes. |
yes | $1290\xf73=430$ |
$5$ | Is last digit is $5$ or $0?$ Yes. | yes | $1290\xf75=258$ |
$10$ | Is last digit $0?$ Yes. | yes | $1290\xf710=129$ |
Thus, $\mathrm{1,290}$ is divisible by $2,3,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.$
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