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Find Multiples and Factors

Identify multiples of 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.

\[\begin{array}{l}1⋅2=2 \\ 2⋅2=4 \\ 3⋅2=6 \\ 4⋅2=8 \\ 5⋅2=10 \\ 6⋅2=12\end{array}\]

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 $3.$

\[\begin{array}{l}1⋅3=3 \\ 2⋅3=6 \\ 3⋅3=9 \\ 4⋅3=12 \\ 5⋅3=15 \\ 6⋅3=18\end{array}\]

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

Counting Number

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$8$

$9$

$10$

$11$

$12$

$\text{Multiples of}\ 2$

$2$

$4$

$6$

$8$

$10$

$12$

$14$

$16$

$18$

$20$

$22$

$24$

$\text{Multiples of}\ 3$

$3$

$6$

$9$

$12$

$15$

$18$

$21$

$24$

$27$

$30$

$33$

$36$

$\text{Multiples of}\ 4$

$4$

$8$

$12$

$16$

$20$

$24$

$28$

$32$

$36$

$40$

$44$

$48$

$\text{Multiples of}\ 5$

$5$

$10$

$15$

$20$

$25$

$30$

$35$

$40$

$45$

$50$

$55$

$60$

$\text{Multiples of}\ 6$

$6$

$12$

$18$

$24$

$30$

$36$

$42$

$48$

$54$

$60$

$66$

$72$

$\text{Multiples of}\ 7$

$7$

$14$

$21$

$28$

$35$

$42$

$49$

$56$

$63$

$70$

$77$

$84$

$\text{Multiples of}\ 8$

$8$

$16$

$24$

$32$

$40$

$48$

$56$

$64$

$72$

$80$

$88$

$96$

$\text{Multiples of}\ 9$

$9$

$18$

$27$

$36$

$45$

$54$

$63$

$72$

$81$

$90$

$99$

$108$

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

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 is either $0,2,4,6,\text{or}\ 8.$ 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 $0,2,4,6,\ \text{or}\ 8,$ then the number is a multiple of $2.$

Example

Try it.

Determine whether each of the following is a multiple of $2\text{:}$

ⓐ $\ 489\$
ⓑ $\ 3,714$

Solution

ⓐ
Is 489 a multiple of 2?
Is the last digit 0, 2, 4, 6, or 8?	No.
	489 is not a multiple of 2.

ⓑ
Is 3,714 a multiple of 2?
Is the last digit 0, 2, 4, 6, or 8?	Yes.
	3,714 is a multiple of 2.

Example

Try it.

Determine whether each of the following is a multiple of $5\text{:}\$

ⓐ $\ 579\$
ⓑ $\ 880$

Solution

ⓐ
Is 579 a multiple of 5?
Is the last digit 5 or 0?	No.
	579 is not a multiple of 5.

ⓑ
Is 880 a multiple of 5?
Is the last digit 5 or 0?	Yes.
	880 is a multiple of 5.

Condensed — the full section is in OpenStax Prealgebra 2e.

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, $375\div 5$ is $75,$ so $375$ is $5⋅75.$ Notice in 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. 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 $0,\ 2,\ 4,\ 6,\ \text{or}\ 8$
$3$	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 $0$

Example

Try it.

Determine whether $1,290$ is divisible by $2,3,5,\text{and}\ 10.$

Solution

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 $0,\ 2,\ 4,\ 6,\ \text{or}\ 8?$ Yes.	yes	$1290\div 2=645$
$3$	$\text{Is sum of digits divisible by}\ 3?$ $1+2+9+0=12$ Yes.	yes	$1290\div 3=430$
$5$	Is last digit $5$ or $0?$ Yes.	yes	$1290\div 5=258$
$10$	Is last digit $0?$ Yes.	yes	$1290\div 10=129$

Thus, $1,290$ is divisible by $2,3,5,\text{and}\ 10.$

Example

Try it.

Determine whether $5,625$ is divisible by $2,3,5,\text{and}\ 10.$

Solution

applies the divisibility tests to $5,625$ and tests the results by finding the quotients.

Divisible by…?	Test	Divisible?	Check
$2$	Is last digit $0,\ 2,\ 4,\ 6,\ \text{or}\ 8?$ No.	no	$5625\div 2=2812.5$
$3$	$\text{Is sum of digits divisible by}\ 3?$ $5+6+2+5=18$ Yes.	yes	$5625\div 3=1875$
$5$	Is last digit is $5$ or $0?$ Yes.	yes	$5625\div 5=1125$
$10$	Is last digit $0?$ No.	no	$5625\div 10=562.5$

Thus, $5,625$ is divisible by $3$ and $5,$ but not $2,$ or $10.$

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 $72=8⋅9$ 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 $24.$ summarizes the different ways that the choreographer can arrange the dancers.

Number of Groups	Dancers per Group	Total Dancers
$1$	$24$	$1⋅24=24$
$2$	$12$	$2⋅12=24$
$3$	$8$	$3⋅8=24$
$4$	$6$	$4⋅6=24$
$6$	$4$	$6⋅4=24$
$8$	$3$	$8⋅3=24$
$12$	$2$	$12⋅2=24$
$24$	$1$	$24⋅1=24$

What patterns do you see in ? 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:

\[1,2,3,4,6,8,12,24\]

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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.

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 $2,3,5,7,11,13,17,\text{and}\ 19.$ 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

Try it.

Identify each number as prime or composite:

ⓐ $\ 83\$
ⓑ $\ 77$

Solution

ⓐ Test each prime, in order, to see if it is a factor of $83$, starting with $2,$ as shown. We will stop when the quotient is smaller than the divisor.

Prime	Test	Factor of $83?$
$2$	Last digit of $83$ is not $0,2,4,6,\text{or}\ 8.$	No.
$3$	$8+3=11,$ and $11$ is not divisible by $3.$	No.
$5$	The last digit of $83$ is not $5$ or $0.$	No.
$7$	$83\div 7=11.857\text{\ldots .}$	No.
$11$	$83\div 11=7.545\text{\ldots }$	No.

We can stop when we get to $11$ because the quotient $\text{(7.545\ldots )}$ is less than the divisor.

We did not find any prime numbers that are factors of $83,$ so we know $83$ is prime.

ⓑ Test each prime, in order, to see if it is a factor of $77.$

Prime	Test	Factor of $77?$
$2$	Last digit is not $0,2,4,6,\text{or}\ 8.$	No.
$3$	$7+7=14,$ and $14$ is not divisible by $3.$	No.
$5$	the last digit is not $5$ or $0.$	No.
$7$	$77\div 7=11$	Yes.

Since $77$ is divisible by $7,$ we know it is not a prime number. It is composite.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

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
4	if the last two digits are a number divisible by 4
5	if the last digit is 5 or 0
6	if divisible by both 2 and 3
10	if the last digit is 0

Factors If $a⋅b=m$, then $a$ and $b$ are factors of $m$, and $m$ is the product of $a$ and $b$.
Find all the factors of a counting number.
1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
  1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
  2. If the quotient is not a counting number, the divisor is not a factor.
2. List all the factor pairs.
3. Write all the factors in order from smallest to largest.
Determine if a number is prime.
1. Test each of the primes, in order, to see if it is a factor of the number.
2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

Find Multiples and Factors

Identify Multiples of Numbers

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

Try it.

$2$

Solution

2, 4, 6, 8, 10 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48

Try it.

$3$

Try it.

$4$

Solution

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

Try it.

$5$

Try it.

$6$

Solution

6, 12, 18, 24, 30, 36, 42, 48

Try it.

$7$

Try it.

$8$

Solution

8, 16, 24, 32, 40, 48

Try it.

$9$

Try it.

$10$

Solution

10, 20, 30, 40

Try it.

$12$

Use Common Divisibility Tests

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

Try it.

$84$

Solution

Divisible by 2, 3, 4, 6

Try it.

$96$

Try it.

$75$

Solution

Divisible by 3, 5

Try it.

$78$

Try it.

$168$

Solution

Divisible by 2, 3, 4, 6

Try it.

$264$

Try it.

$900$

Solution

Divisible by 2, 3, 4, 5, 6, 10

Try it.

$800$

Try it.

$896$

Solution

Divisible by 2, 4

Try it.

$942$

Try it.

$375$

Solution

Divisible by 3, 5

Try it.

$750$

Try it.

$350$

Solution

Divisible by 2, 5, 10

Try it.

$550$

Try it.

$1430$

Solution

Divisible by 2, 5, 10

Try it.

$1080$

Try it.

$22,335$

Solution

Divisible by 3, 5

Try it.

$39,075$

Find All the Factors of a Number

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

Try it.

$36$

Solution

1, 2, 3, 4, 6, 9, 12, 18, 36

Try it.

$42$

Try it.

$60$

Solution

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Try it.

$48$

Try it.

$144$

Solution

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72,144

Try it.

$200$

Try it.

$588$

Solution

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588

Try it.

$576$

Identify Prime and Composite Numbers

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

Try it.

$43$

Solution

prime

Try it.

$67$

Try it.

$39$

Solution

composite

Try it.

$53$

Try it.

$71$

Solution

prime

Try it.

$119$

Try it.

$481$

Solution

composite

Try it.

$221$

Try it.

$209$

Solution

composite

Try it.

$359$

Try it.

$667$

Solution

composite

Try it.

$1771$

Try it.

If a number is divisible by $2$ and by $3,$ why is it also divisible by $6?$

Try it.

What is the difference between prime numbers and composite numbers?

ⓐ 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?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (30)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Which of the following numbers are counting numbers (natural numbers)?
$0,4,215$
If you missed this problem, review .

Reveal the answer

$4$ and $215$
Find the sum of $3,5,$ and $7.$
If you missed the problem, review .

Reveal the answer

$15$
Determine whether each of the following is a multiple of $2\text{:}$
1. ⓐ $\ 489\$
2. ⓑ $\ 3,714$
Reveal the answer

ⓐ
Is 489 a multiple of 2?
Is the last digit 0, 2, 4, 6, or 8? No.
489 is not a multiple of 2.
ⓑ
Is 3,714 a multiple of 2?
Is the last digit 0, 2, 4, 6, or 8? Yes.
3,714 is a multiple of 2.
Determine whether each number is a multiple of $2\text{:}$
1. ⓐ $\ 678\$
2. ⓑ $\ 21,493$
Reveal the answer
1. ⓐ yes
2. ⓑ no
Determine whether each number is a multiple of $2\text{:}$
1. ⓐ $\ 979\$
2. ⓑ $\ 17,780$
Reveal the answer
1. ⓐ no
2. ⓑ yes
Determine whether each of the following is a multiple of $5\text{:}\$
1. ⓐ $\ 579\$
2. ⓑ $\ 880$
Reveal the answer

ⓐ
Is 579 a multiple of 5?
Is the last digit 5 or 0? No.
579 is not a multiple of 5.
ⓑ
Is 880 a multiple of 5?
Is the last digit 5 or 0? Yes.
880 is a multiple of 5.
Determine whether each number is a multiple of $5.$
1. ⓐ $\ 675\$
2. ⓑ $\ 1,578$
Reveal the answer
1. ⓐ yes
2. ⓑ no
Determine whether each number is a multiple of $5.$
1. ⓐ $\ 421\$
2. ⓑ $\ 2,690$
Reveal the answer
1. ⓐ no
2. ⓑ yes
Determine whether each of the following is a multiple of $10\text{:}\$
1. ⓐ $\ 425\$
2. ⓑ $\ 350$
Reveal the answer

ⓐ
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. ⓐ $\ 179\$
2. ⓑ $\ 3,540$
Reveal the answer
1. ⓐ no
2. ⓑ yes
Determine whether each number is a multiple of $10\text{:}$
1. ⓐ $\ 110\$
2. ⓑ $\ 7,595$
Reveal the answer
1. ⓐ yes
2. ⓑ no

Determine whether each of the given numbers is a multiple of $3\text{:}\$

ⓐ $\ 645\$
ⓑ $\ 10,519$

Reveal the answer

ⓐ 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.	$645\div 3$
The quotient is 215.	$3⋅215=645$

ⓑ 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..	$645\div 3$
We can check this by dividing by 10,519 by 3.	$\begin{array}{l}3,506\text{R}1 \\ 310,519\ \end{array}$

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. ⓐ $\ 954\$
2. ⓑ $\ 3,742$
Reveal the answer
1. ⓐ yes
2. ⓑ no
Determine whether each number is a multiple of $3\text{:}$
1. ⓐ $\ 643\$
2. ⓑ $\ 8,379$
Reveal the answer
1. ⓐ no
2. ⓑ yes

Determine whether $1,290$ is divisible by $2,3,5,\text{and}\ 10.$

Reveal the answer

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 $0,\ 2,\ 4,\ 6,\ \text{or}\ 8?$ Yes.	yes	$1290\div 2=645$
$3$	$\text{Is sum of digits divisible by}\ 3?$ $1+2+9+0=12$ Yes.	yes	$1290\div 3=430$
$5$	Is last digit $5$ or $0?$ Yes.	yes	$1290\div 5=258$
$10$	Is last digit $0?$ Yes.	yes	$1290\div 10=129$

Thus, $1,290$ is divisible by $2,3,5,\text{and}\ 10.$

Determine whether the given number is divisible by $2,3,5,\text{and}\ 10.$
$6240$

Reveal the answer

Divisible by 2, 3, 5, and 10
Determine whether the given number is divisible by $2,3,5,\text{and}\ 10.$
$7248$

Reveal the answer

Divisible by 2 and 3, not 5 or 10.

Determine whether $5,625$ is divisible by $2,3,5,\text{and}\ 10.$

Reveal the answer

applies the divisibility tests to $5,625$ and tests the results by finding the quotients.

Divisible by…?	Test	Divisible?	Check
$2$	Is last digit $0,\ 2,\ 4,\ 6,\ \text{or}\ 8?$ No.	no	$5625\div 2=2812.5$
$3$	$\text{Is sum of digits divisible by}\ 3?$ $5+6+2+5=18$ Yes.	yes	$5625\div 3=1875$
$5$	Is last digit is $5$ or $0?$ Yes.	yes	$5625\div 5=1125$
$10$	Is last digit $0?$ No.	no	$5625\div 10=562.5$

Thus, $5,625$ is divisible by $3$ and $5,$ but not $2,$ or $10.$

Determine whether the given number is divisible $\text{by}\ 2,3,5,\text{and}\ 10.$
$4962$

Reveal the answer

Divisible by 2 and 3, not 5 or 10.
Determine whether the given number is divisible $\text{by}\ 2,3,5,\text{and}\ 10.$
$3765$

Reveal the answer

Divisible by 3 and 5.
Find all the factors of $72.$

Reveal the answer

Divide $72$ by each of the counting numbers starting with $1.$ If the quotient is a whole number, the divisor and quotient are a pair of factors.

The next line would have a divisor of $9$ and a quotient of $8.$ The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to greatest, we have
$1,2,3,4,6,8,9,12,18,24,36,\text{and}\ 72$
Find all the factors of the given number:
$96$

Reveal the answer

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Find all the factors of the given number:
$80$

Reveal the answer

1, 2, 4, 5, 8, 10, 16, 20, 40, 80

Identify each number as prime or composite:

ⓐ $\ 83\$
ⓑ $\ 77$

Reveal the answer

ⓐ Test each prime, in order, to see if it is a factor of $83$, starting with $2,$ as shown. We will stop when the quotient is smaller than the divisor.

Prime	Test	Factor of $83?$
$2$	Last digit of $83$ is not $0,2,4,6,\text{or}\ 8.$	No.
$3$	$8+3=11,$ and $11$ is not divisible by $3.$	No.
$5$	The last digit of $83$ is not $5$ or $0.$	No.
$7$	$83\div 7=11.857\text{\ldots .}$	No.
$11$	$83\div 11=7.545\text{\ldots }$	No.

We can stop when we get to $11$ because the quotient $\text{(7.545\ldots )}$ is less than the divisor.

We did not find any prime numbers that are factors of $83,$ so we know $83$ is prime.

ⓑ Test each prime, in order, to see if it is a factor of $77.$

Prime	Test	Factor of $77?$
$2$	Last digit is not $0,2,4,6,\text{or}\ 8.$	No.
$3$	$7+7=14,$ and $14$ is not divisible by $3.$	No.
$5$	the last digit is not $5$ or $0.$	No.
$7$	$77\div 7=11$	Yes.

Since $77$ is divisible by $7,$ we know it is not a prime number. It is composite.

Identify the number as prime or composite:
$91$

Reveal the answer

composite
Identify the number as prime or composite:
$137$

Reveal the answer

prime

Banking Frank’s grandmother gave him $\text{\$100}$ at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $\text{\$15}$ to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after graduation	Total number of dollars Frank put in the account	Simplified Total
$0$	$100$	$100$
$1$	$100+15$	$115$
$2$	$100+15⋅2$	$130$
$3$	$100+15⋅3$
$4$	$100+15⋅[\ ]$
$5$	$100+[\ ]$
$6$
$20$
$x$

Reveal the answer

Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $\text{\$75}$ to open the account. Every week, she added $\text{\$20}$ to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after opening the account	Total number of dollars Gina put in the account	Simplified Total
$0$	$75$	$75$
$1$	$75+20$	$95$
$2$	$75+20⋅2$	$115$
$3$	$75+20⋅3$
$4$	$75+20⋅[\ ]$
$5$	$75+[\ ]$
$6$
$20$
$x$

If a number is divisible by $2$ and by $3,$ why is it also divisible by $6?$
What is the difference between prime numbers and composite numbers?

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Find Multiples and Factors

Identify multiples of numbers
Use common divisibility tests
Find all the factors of a number
Identify prime and composite numbers

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Try your own

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

More in Arithmetic

Order of operations Fractions Percentages Powers and roots

Divisibility Tests
A number is divisible by
\(2\)	if the last digit is \(0,\ 2,\ 4,\ 6,\ \text{or}\ 8\)
\(3\)	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 \(0\)

Divisible by…?	Test	Divisible?	Check
\(2\)	Is last digit \(0,\ 2,\ 4,\ 6,\ \text{or}\ 8?\) Yes.	yes	\(1290\div 2=645\)
\(3\)	\(\text{Is sum of digits divisible by}\ 3?\) \(1+2+9+0=12\) Yes.	yes	\(1290\div 3=430\)
\(5\)	Is last digit \(5\) or \(0?\) Yes.	yes	\(1290\div 5=258\)
\(10\)	Is last digit \(0?\) Yes.	yes	\(1290\div 10=129\)

Divisible by…?	Test	Divisible?	Check
\(2\)	Is last digit \(0,\ 2,\ 4,\ 6,\ \text{or}\ 8?\) No.	no	\(5625\div 2=2812.5\)
\(3\)	\(\text{Is sum of digits divisible by}\ 3?\) \(5+6+2+5=18\) Yes.	yes	\(5625\div 3=1875\)
\(5\)	Is last digit is \(5\) or \(0?\) Yes.	yes	\(5625\div 5=1125\)
\(10\)	Is last digit \(0?\) No.	no	\(5625\div 10=562.5\)

Number of Groups	Dancers per Group	Total Dancers
\(1\)	\(24\)	\(1⋅24=24\)
\(2\)	\(12\)	\(2⋅12=24\)
\(3\)	\(8\)	\(3⋅8=24\)
\(4\)	\(6\)	\(4⋅6=24\)
\(6\)	\(4\)	\(6⋅4=24\)
\(8\)	\(3\)	\(8⋅3=24\)
\(12\)	\(2\)	\(12⋅2=24\)
\(24\)	\(1\)	\(24⋅1=24\)

Prime	Test	Factor of \(83?\)
\(2\)	Last digit of \(83\) is not \(0,2,4,6,\text{or}\ 8.\)	No.
\(3\)	\(8+3=11,\) and \(11\) is not divisible by \(3.\)	No.
\(5\)	The last digit of \(83\) is not \(5\) or \(0.\)	No.
\(7\)	\(83\div 7=11.857\text{\ldots .}\)	No.
\(11\)	\(83\div 11=7.545\text{\ldots }\)	No.

Prime	Test	Factor of \(77?\)
\(2\)	Last digit is not \(0,2,4,6,\text{or}\ 8.\)	No.
\(3\)	\(7+7=14,\) and \(14\) is not divisible by \(3.\)	No.
\(5\)	the last digit is not \(5\) or \(0.\)	No.
\(7\)	\(77\div 7=11\)	Yes.

ⓐ
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.

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.	\(645\div 3\)
The quotient is 215.	\(3⋅215=645\)

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..	\(645\div 3\)
We can check this by dividing by 10,519 by 3.	\(\begin{array}{l}3,506\text{R}1 \\ 310,519\ \end{array}\)