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Properties of Multiplication
The properties of multiplication are certain rules that are used while multiplying numbers. These properties help in simplifying expressions easily and hence, have a significant role in solving all kinds of mathematical expressions, whether they are algebraic expressions, fractions, or integers. This article gives an insight into the different types of properties of multiplication.
What are the Properties of Multiplication?
The properties of multiplication are those features that are used when we multiply two or more numbers in an expression. The different properties of multiplication have various types of rules as explained in the following sections.
Associative Property of Multiplication
According to the Associative property of multiplication, changing the grouping of numbers does not have an effect on the product of numbers. For example, (4 × 6) × 3 = 4 × (6 × 3) = 72. The formula that is used to express this property is, (a × b) × c = a × (b × c)
Commutative Property of Multiplication
The Commutative property of multiplication states that any change in the order of the factors does not affect the product. For example, 3 × 5 × 2 = 2 × 3 × 5 = 30. The formula of the commutative property of multiplication is expressed as, a × b = b × a
Distributive Property of Multiplication
The Distributive property of multiplication is applied to addition and subtraction. It is represented as, a(b + c) = ab + ac; and a(b  c) = ab  ac. According to this property, when a number is multiplied by the sum of two or more addends given in brackets, we can solve it by multiplying this number to both the addends individually, and then their products are added together. This product is the same if we multiply the number by the sum of the two addends. For example, let us solve 5(2 + 4) using the usual rules of simplification where we first solve the brackets, and then we multiply the number with the result. This means, 5(2 + 4) = 5 × 6 = 30. Now, when we apply the distributive property of multiplication, we will multiply the number outside the bracket with the first addend inside the brackets, and then multiply the number with the second addend inside the bracket. This means, 5(2 + 4) = (5 × 2) + (5 × 4) = 10 + 20 = 30. We can see that the result is the same. It should be noted that the Distributive property of multiplication is applied in the same way in the case of subtraction.
Identity Property of Multiplication
The Identity property of multiplication which is also known as the Multiplicative Identity Property states that when a number is multiplied by 1, the product is always the number itself. It is represented as, a × 1 = a. For example, 5 × 1 = 5, or 1 × 17 = 17.
Zero Property of Multiplication
According to the Zero property of multiplication, when a number is multiplied with 0, the product is always 0. It is represented as, a × 0 = 0. For example, 42 × 0 = 0, or 0 × 23 = 0.
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Examples of Properties of Multiplication

Example 1: Which statement is an example of the Identity property of multiplication?
a.) 98 × 1 = 98
b.) 5 × 7 = 35
c.) 5 × 4 = 4 × 25
d.) (9 × 8) × 7 = 9 × (8 × 7)
Solution: Using the properties of multiplication, we can say that option (a.) 98 × 1 = 98 is an example of the Identity property of multiplication because when 98 is multiplied with 1, it results in the number 98 itself.

Example 2: Use the properties of multiplication to fill in the missing number: 435 × 56 × 12 = 12 × ___ × 56
Solution: According to the Commutative property of multiplication, we can conclude that the missing number is 435.

Example 3: Fill in the missing number using the properties of multiplication: (456 × 212) × 10 = 456 × (___ × 10)
Solution: Using the Associative property of multiplication, we can say that the missing number is 212.
FAQs on Properties of Multiplication
What are the Properties of Multiplication?
The properties of multiplication are those set of rules that help in simplifying expressions. There are 5 basic properties of multiplication.
 The Associative property of multiplication: (a × b) × c = a × (b × c)
 The Commutative property of multiplication: a × b = b × a
 The Identity property of multiplication: a × 1 = a
 The Distributive property of multiplication: a(b + c) = ab + ac; and a(b  c) = ab  ac
 The Zero property of multiplication. a × 0 = 0
What is the Distributive Property of Multiplication over Addition?
The Distributive property of multiplication over addition means that when a number is multiplied with the sum of two or more addends, it will give the same result if we multiply each addend separately by the number given outside the brackets. For example, let us solve 10(5 + 8). If we solve it in the usual manner, we get 10 × 13 = 130. Now, if we apply the distributive property, we will multiply 10 with 5 and 8 individually and then add their products together. This will result in, 10(5 + 8) = (10 × 5) + (10 × 8) = 50 + 80 = 130.
How do we Apply the Properties of Multiplication?
The properties of multiplication can be applied when we multiply integers, fractions, decimals or even algebraic expressions. For example, the identity property of multiplication says that any number multiplied by 1 results in the number itself. Similarly, the Associative property of multiplication tells us that changing the grouping of numbers does not have an impact on the product of the numbers. In the same way all the other properties of multiplication can be applied to make calculations easier.
What is the Difference Between the Commutative and Associative Properties of Multiplication?
The Commutative property of multiplication says that if we change the order of the factors of a number, the product remains the same. For example, 7 × 20 = 20 × 7 = 140. The Associative property of multiplication states that if the grouping of a set of numbers is changed, the product still remains the same. For example, 22 × (4 × 10) = (22 × 4) × 10 = 880.
How many Properties of Multiplication are there?
There are five basic properties of multiplication  the Associative property of multiplication, the Commutative property of multiplication, the Identity property of multiplication, the Distributive property of multiplication, and the Zero property of multiplication. Each of them has its unique features which helps in simplifying expressions easily.
Give an Example of the Associative Property of Multiplication.
The Associative property of multiplication can be applied to many expressions. For example, if we group a set of numbers with brackets and write them as (102 × 50) × 20, we get the product of these numbers as 102000. Now, if we group these numbers as, 102 × (50 × 20), we get the same product 102000. Therefore, the Associative property of multiplication can be written as, (a × b) × c = a × (b × c).
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