Multiplying Whole Numbers: Properties of Multiplication

Learning Outcomes

Identify and use the multiplication property of zero
Identify and use the identity property of multiplication
Identify and use the commutative property of multiplication
Multiply multiple-digit whole numbers using columns that represent place value

Multiply Whole Numbers

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.

Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers. Start with memorizing the multiplication facts through $9 \times 9$ . Knowing the times tables up to $12 \times 12$ will allow your brain to focus on problem solving in future math questions so you’re not stuck on the arithmetic.

$×$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$1$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$2$	$0$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$	$20$	$22$	$24$
$3$	$0$	$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$	$27$	$30$	$33$	$36$
$4$	$0$	$4$	$8$	$12$	$16$	$20$	$24$	$28$	$32$	$36$	$40$	$44$	$48$
$5$	$0$	$5$	$10$	$15$	$20$	$25$	$30$	$35$	$40$	$45$	$50$	$55$	$60$
$6$	$0$	$6$	$12$	$18$	$24$	$30$	$36$	$42$	$48$	$54$	$60$	$66$	$72$
$7$	$0$	$7$	$14$	$21$	$28$	$35$	$42$	$49$	$56$	$63$	$70$	$77$	$84$
$8$	$0$	$8$	$16$	$24$	$32$	$40$	$48$	$56$	$64$	$72$	$80$	$88$	$96$
$9$	$0$	$9$	$18$	$27$	$36$	$45$	$54$	$63$	$72$	$81$	$90$	$99$	$108$
$10$	$0$	$10$	$20$	$30$	$40$	$50$	$60$	$70$	$80$	$90$	$100$	$110$	$120$
$11$	$0$	$11$	$22$	$33$	$44$	$55$	$66$	$77$	$88$	$99$	$110$	$121$	$132$
$12$	$0$	$12$	$24$	$36$	$48$	$60$	$72$	$84$	$96$	$108$	$120$	$132$	$144$

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.

Multiplication Property of Zero

The product of any number and $0$ is $0$ .

$\begin{array}{}\\ a\cdot 0=0\hfill \\ 0\cdot a=0\end{array}$

example

Multiply:

$0\cdot 11$
$\left(42\right)0$

Solution:

1.	$0\cdot 11$
The product of any number and zero is zero.	$0$
2.	$\left(42\right)0$
Multiplying by zero results in zero.	$0$

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What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and $1$ is called the multiplicative identity.

Identity Property of Multiplication

The product of any number and $1$ is the number.

$\begin{array}{c}1\cdot a=a\\ a\cdot 1=a\end{array}$

example

Multiply:

$\left(11\right)1$
$1\cdot 42$

Show Solution

Solution:

1.	$\left(11\right)1$
The product of any number and one is the number.	$11$
2.	$1\cdot 42$
Multiplying by one does not change the value.	$42$

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Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that $8+9=17$ is the same as $9+8=17$ .

Is this also true for multiplication? Let’s look at a few pairs of factors.

$4\cdot 7=28\quad 7\cdot 4=28$
$9\cdot 7=63\quad 7\cdot 9=63$
$8\cdot 9=72\quad 9\cdot 8=72$

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

Commutative Property of Multiplication

Changing the order of the factors does not change their product.

$a\cdot b=b\cdot a$

example

Multiply:

$8\cdot 7$
$7\cdot 8$

Show Solution

Solution:

1.	$8\cdot 7$
Multiply.	$56$
2.	$7\cdot 8$
Multiply.	$56$

Changing the order of the factors does not change the product.

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To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

$\begin{array}{c}\hfill 27\\ \hfill \underset{\text{___}}{\times 3}\end{array}$

We start by multiplying $3$ by $7$ .

$3\times 7=21$

We write the $1$ in the ones place of the product. We carry the $2$ tens by writing $2$ above the tens place.

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Then we multiply the $3$ by the $2$ , and add the $2$ above the tens place to the product. So $3\times 2=6$ , and $6+2=8$ . Write the $8$ in the tens place of the product.

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The product is $81$ .

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.

example

Multiply: $15\cdot 4$

Show Solution

Solution

Write the numbers so the digits $5$ and $4$ line up vertically.	$\begin{array}{c}\hfill 15\\ \hfill \underset{\text{_____}}{\times 4}\end{array}$
Multiply $4$ by the digit in the ones place of $15$ . $4\cdot 5=20$ .
Write $0$ in the ones place of the product and carry the $2$ tens.	$\begin{array}{c}\hfill \stackrel{2}{1}5\\ \hfill \underset{\text{_____}}{\times 4}\\ \hfill 0\end{array}$
Multiply $4$ by the digit in the tens place of $15$ . $4\cdot 1=4$ .Add the $2$ tens we carried. $4+2=6$ .
Write the $6$ in the tens place of the product.	$\begin{array}{c}\hfill \stackrel{2}{1}5\\ \hfill \underset{\text{_____}}{\times 4}\\ \hfill 60\end{array}$

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https://ohm.lumenlearning.com/multiembedq2.php?id=144427&theme=oea&iframe_resize_id=mom

example

Multiply: $286\cdot 5$

Show Solution

Solution

Write the numbers so the digits $5$ and $6$ line up vertically.	$\begin{array}{c}\hfill 286\\ \hfill \underset{\text{_____}}{\times 5}\end{array}$
Multiply $5$ by the digit in the ones place of $286$ . $5\cdot 6=30$
Write the $0$ in the ones place of the product and carry the $3$ to the tens place.Multiply $5$ by the digit in the tens place of $286$ . $5\cdot 8=40$	$\begin{array}{}\\ \hfill 2\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 0\end{array}$
Add the $3$ tens we carried to get $40+3=43$ .Write the $3$ in the tens place of the product and carry the $4$ to the hundreds place.	$\begin{array}{c}\hfill \stackrel{4}{2}\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 30\end{array}$
Multiply $5$ by the digit in the hundreds place of $286$ . $5\cdot 2=10$ .Add the $4$ hundreds we carried to get $10+4=14$ . Write the $4$ in the hundreds place of the product and the $1$ to the thousands place.	$\begin{array}{c}\hfill \stackrel{4}{2}\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 1,430\end{array}$

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When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.

Multiply two whole numbers to find the product

Write the numbers so each place value lines up vertically.
Multiply the digits in each place value.
- Work from right to left, starting with the ones place in the bottom number.
  - Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
  - If a product in a place value is more than $9$ , carry to the next place value.
  - Write the partial products, lining up the digits in the place values with the numbers above.
- Repeat for the tens place in the bottom number, the hundreds place, and so on.
- Insert a zero as a placeholder with each additional partial product.
Add the partial products.

example

Multiply: $62\left(87\right)$

Show Solution

Solution

Write the numbers so each place lines up vertically.
Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62. $7\cdot 2=14$ . Write the 4 in the ones place of the product and carry the 1 to the tens place.
Multiply 7 by the digit in the tens place of 62. $7\cdot 6=42$ . Add the 1 ten we carried. $42+1=43$ latex]. Write the 3 in the tens place of the product and the 4 in the hundreds place.
The first partial product is $434$ .
Now, write a $0$ under the $4$ in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of $87$ by $62$ .Multiply $8$ by the digit in the ones place of $62$ $8\cdot 2=16$ . Write the $6$ in the next place of the product, which is the tens place. Carry the $1$ to the tens place.
Multiply $8$ by $6$ , the digit in the tens place of $62$ , then add the $1$ ten we carried to get $49$ .Write the $9$ in the hundreds place of the product and the $4$ in the thousands place.
The second partial product is $4960$ . Add the partial products.

The product is $5,394$ .

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example

Multiply:

$47\cdot 10$
$47\cdot 100$

Show Solution

Solution

$47\cdot 10$	$\begin{array}{c}\hfill 47\\ \hfill \underset{\text{___}}{\times 10}\\ \hfill 00\\ \hfill \underset{\text{___}}{470}\\ \hfill 470\end{array}$
$47\cdot 100$	$\begin{array}{c}\hfill 47\\ \hfill \underset{\text{_____}}{\times 100}\\ \hfill 00\\ \hfill \underset{\text{_____}}{\begin{array}{c}\hfill 000\\ \hfill 4700\end{array}}\\ \hfill 4,700\end{array}$

When we multiplied $47$ times $10$ , the product was $470$ . Notice that $10$ has one zero, and we put one zero after $47$ to get the product. When we multiplied $47$ times $100$ , the product was $4,700$ . Notice that $100$ has two zeros and we put two zeros after $47$ to get the product.

Do you see the pattern? If we multiplied $47$ times $10,000$ , which has four zeros, we would put four zeros after $47$ to get the product $470,000$ .

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Multiply:

https://ohm.lumenlearning.com/multiembedq2.php?id=144433&theme=oea&iframe_resize_id=mom7

example

Multiply: $\left(354\right)\left(438\right)$

Show Solution

Solution
There are three digits in the factors so there will be $3$ partial products. We do not have to write the $0$ as a placeholder as long as we write each partial product in the correct place.

An image of the multiplication problem

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example

Multiply: $\left(896\right)201$

Show Solution

Solution
There should be $3$ partial products. The second partial product will be the result of multiplying $896$ by $0$ .

An image of the multiplication problem
Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the $2$ in the hundreds place, as shown.
Multiply by $10$ , but insert only one zero as a placeholder in the tens place. Multiply by $200$ , putting the $2$ from the $12$ . $2\cdot 6=12$ in the hundreds place.

$\begin{array}{}\\ \\ \hfill 896\\ \hfill \underset{\text{_____}}{\times 201}\\ \hfill 896\\ \hfill \underset{\text{__________}}{17920}\\ \hfill 180,096\end{array}$

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When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

to multiply	$8\cdot 3\cdot 2$
first multiply $8\cdot 3$	$24\cdot 2$
then multiply $24\cdot 2$	$48$

In the video below, we summarize the concepts presented on this page including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.

Module 1: Historical Counting Systems

Learning Outcomes

Multiply Whole Numbers

Multiplication Property of Zero

example

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Identity Property of Multiplication

example

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Commutative Property of Multiplication

example

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Multiply two whole numbers to find the product

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Candela Citations