Properties of Multiplication
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Last updated on December 11, 2025

Multiplication is the process of adding a number repeatedly. The numbers that are being multiplied together are called factors. The result of multiplying two or more numbers is called the product. In math, multiplication is one of the basic arithmetic operations that we use in our daily lives.

What is Multiplication?

Multiplication is used when we have to multiply two or more numbers to find the product. Let us learn more about multiplication in this article.

If there are 4 packets of ice cream and each box has 6 ice creams, find the total number of ice creams.

Solution: We can solve this question by adding 6 + 6 + 6 + 6 = 24.

But it takes a lot of time and effort, so we solve it as 6 × 4 = 24. In other words, multiplication is the repeated addition of numbers.

What are the Properties of Multiplication?

The properties of multiplication are special rules that help us understand how numbers behave when we multiply them. The main multiplication properties are: Commutative, Associative, Distributive, Identity, and Zero Property.

Commutative Property a × b = b ×a Associative Property (a × b) × c = a × (b ×c) Distributive Property a(b + c) = ab + ac
a(b - c) = ab - ac Identity Property a × 1 = a
1 × a = a Zero Property

a × 0 = 0

0 × a = 0

Commutative Property: This property states that when we multiply two numbers, changing their order doesn't affect the product. Multiplying 3 rows by 4 columns or 4 columns by 3 rows gives the same result:

3 × 4 = 4 × 3 = 12.

Associative Property: This property states that if we multiply three or more numbers one after the other, the order in which we group them does not affect the product. For example, let us take 2, 4, and 3.

If we multiply them in the same order, we get: (2 × 4) × 3 = 24(2 × 4) × 3 = 24.

If we rearrange the order and multiply: 3 × (2 × 4) = 24 

(2 × 4) = 8, then 8 × 3 = 24

Distributive Property: According to this property, if we multiply a number by a sum, the result will be equal to another expression where we multiply the same number by each addend separately and then add the products. Let's see this with an example:

First multiply 2 × 4 = 8, then 2 × 2 = 4, finally add 8 + 4 = 12.

2 × (4 + 2) = (2 × 4) + (2 × 2) = 8 + 4 = 12

Identity Property of Multiplication: This property states that when we multiply any number by 1, the product stays the same. It is also called the Multiplicative Identity Property. In symbols, a × 1 = a.

For example: 2 × 1 = 2 and 1 × 25 = 25.

Zero Property of Multiplication: Multiplying any number by 0 always results in 0. It can be written as a × 0 = 0.

For example: 15 × 0 = 0 and 0 × 33 = 0.

Other Important Properties of Multiplication

Multiplication has essential properties that help us understand the behavior of numbers in various situations. The properties are the closure property, the multiplication property of equality, and the inverse property of multiplication.

Closure Property of Multiplication: The closure property of multiplication states that the product of multiplying two numbers from a set will also belong to that same set. In other words, if a and b are whole numbers, then a × b will always be a whole number. 

For example: 6 × 5 = 30

-5 × 5 = -25

Multiplication Property of Equality: The multiplicative property of equality states that if you multiply both sides of an equation by the same nonzero number, the equality stays the same. It can be represented as: if a = b, then a × c = b × c, where c is any nonzero number.

For example, \({1\over4}{x} = {100}\)

Multiplying both sides by 4: 

\({4} × {1 \over 4}{x} = {100 × 4} \)

x = 400

Inverse Property of Multiplication: The Inverse Property of Multiplication states that when a number is multiplied by its reciprocal, the product is always 1. If a ≠ 0, then: a × (1/a) = 1.

For example, \(5 × {1\over 5} = 1 \)

\({5\over 2} × {2\over 5} = 1 \)

\( {-4} × {-}{1\over 4} = 1\)

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Tips and Tricks to master Properties of Multiplication

Understanding the properties of multiplication establishes a solid base for number sense and algebraic reasoning. There are numerous tips and tricks to help students create fluency and accuracy when using multiplication properties:
 

  • Identify Key Properties: Know the key properties: commutative, associative, distributive, identity, and zero, in order to apply properly to solve mathematical problems.
     
  • Use Variable Representation: Represent the properties with symbols (for example, a × b = b × a) to support algebraic knowledge and logical reasoning.
     
  • Examine Patterns in Numbers: Recognize similarities while operating with numbers through multiplication, that will solidify conceptual understanding while simultaneously providing support for students when calculating mentally.
     
  • Combine Properties when Simplifying: Incorporate multiple properties during multiplication when simplifying a complex expression involving whole numbers and/or algebraic properties to represent the information in a simpler form.
     
  • Test Examples to Verify Properties: Verify a new property by using a different set of numbers through a separate example, to confirm both practical use and conceptual understanding.
     
  • Use Real-Life Examples: Parents can connect multiplication properties to everyday situations, such as arranging chairs (commutative), grouping toys (associative), or distributing snacks (distributive). 
     
  • Use Color-Coding: Parents and teachers can highlight different parts of an expression in various colors (like separating addends in the distributive property).
     
  • Visual Representation: Teachers can use number lines, arrays, counters, or area models to show how properties work.

Common Mistakes and How to Avoid Them in Properties of Multiplication

Learning the properties of multiplication is essential for solving problems accurately. However, students often make mistakes due to misapplication or misunderstanding of these properties. In this section we will learn some common mistakes and ways to avoid them to mastery in multiplication.

Real-life Applications of Properties of Multiplication

Properties of multiplication are often applied to practical calculations in everyday life. Some of them are listed below:

  • When you're shopping, the order in which you multiply the number of items and the price doesn’t matter. For example, if you buy 3 notebooks for ₹50 each = ₹150. Its reverse, i.e., 50 notebooks for ₹3 each = ₹150. The total cost will be the same!
  • When organizing a group of people into smaller teams, how you group them doesn’t change the total number of people. For example, if you have 2 groups of 3 people, and each group is made of 4 smaller teams, you can multiply in any order (2 × 3 × 4 = 24) to find the total number of people.
  • When calculating a total cost, you can break it down into parts to make it easier. For example, if a shirt costs ₹200 and pants cost ₹300, you can use the distributive property to find the total cost for 2 sets of clothes:
    2 × (200 + 300) = (2 × 200) + (2 × 300) = ₹400 + ₹600 = ₹1000.
  • Multiplying any number by 1 leaves it unchanged. For example, if you’re copying 1 document 5 times, the total number of copies is still just 5 (1 × 5 = 5). It helps in counting or keeping the original quantity unchanged.
  • Multiplying any number by 0 gives 0. For example, if there are 0 apples, and you want to multiply them by 5 (0 × 5), you’ll still have 0 apples — nothing changes.

What Are Numbers? 🔢 | Fun Explanation with 🎯 Real-Life Examples for Kids | ✨BrightCHAMPS Math

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Problem 1

What is 6 × 9 and 9 × 6

Okay, lets begin

6 × 9 = 54 and 9 × 6 = 54.

Explanation

The product we get after multiplying a number with another number is the same as the product that we get after changing the order of the multiplier and the multiplicand. 9 × 6 is the same as 6 × 9 in terms of the final answer (product).
 

Well explained 👍

Problem 2

What is (3 × 4) × 2 and 3 × (2 × 4)?

Okay, lets begin

24. 
 

Explanation

(3 × 4) × 2 = 12 × 2 = 24

3 × (2 × 4) = 3 × 8 = 24

They both are giving the same answer, 24.

That is because if we are changing the grouping of numbers, we still get the same answer.

This means, changing the group of numbers doesn't change the product.

Well explained 👍

Problem 3

What is 71 × 1?

Okay, lets begin

 71.
 

Explanation

If any number ‘n’ is multiplied by 1 the answer will be the number ‘n’ itself.
 

Well explained 👍

Problem 4

What is 450 × 0?

Okay, lets begin

 0.

Explanation

When we multiply any number by 0, the result will be 0 only.
 

Well explained 👍

Problem 5

Use the distributive property to calculate 3 × (6 + 2).

Okay, lets begin

24.
 

Explanation

 (3 × 6) + (3 × 2) = 18 + 6 = 24

According to the distributive property, 3 can be multiplied with 6 and 2 separately, then add the products to get the final answer. 

Well explained 👍

FAQs for Properties of Multiplication

1.What are the 5 properties of multiplication?

The properties of multiplication are commutative, associative, distributive, identity, and zero. These properties help in easy calculations and to solve problems efficiently. 
 

2.What is the associative property of multiplication?

The associative property of multiplication says when multiplying three or more numbers, the way you group them does not change the product. In other words, you can change the parentheses, and the answer stays the same.

3.What is the identity property of multiplication?

The identity property of multiplication says any number multiplied by 1 is equal to the number itself. That’s because the number keeps its identity and does not change.

4.What is a distributive property of multiplication?

The distributive property of multiplication tells us that when we multiply a number by a sum or difference, we can multiply the number by each part separately, and then add or subtract the results.

5.Are 3 and 8 multiples of 24? Is 3 a multiple of 24? Can we write 3 = 24 × n for some whole number n?

No, because 24 × 1 = 24, and that’s already bigger than 3. So 3 is not a multiple of 24. The same logic applies to 8 as well. Therefore, 3 and 8 are not multiples of 24.
 

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.