Question

Identify the property used in the following: $2 \times 13 + 8 \times 13 = \left( {2 + 8} \right) \times 13$
A.Commutative
B.Closure
C.Associative
D.Distributive

Answer 1

Hint: Here, we will first find the form of the equation. Then we will analyze each property given in the question and compare it with the given equation to find the required answer. A property is a characteristic of a set.

Complete step-by-step answer:
The given equation is $2 \times 13 + 8 \times 13 = \left( {2 + 8} \right) \times 13$
It is of the form $\left( {a + b} \right) \times c = a \times c + b \times c$
We will now analyze all the terms in the given options to find the equation that satisfies which property.
Commutative Property is a property in which the number on which we perform the operation can be swapped from their position without making a difference in any answer.
Commutative Property is denoted as $a + b = b + a$.
Commutative Property is true for Addition and Multiplication but not for Subtraction and Division. So, the given expression satisfies the distributive property using the addition or distributive property of Multiplication over Addition.
Closure Property is a property in which a set of numbers from a set in the operation results in the number from the same set. Closure Property is denoted as $a,b \in {\mathbf{Z}}$, then $a + b \in {\mathbf{Z}}$ .
Associative Property is a property in which three or more numbers are added, the sum is the same irrespective of the grouping of Addends.
 Associative Property is denoted as $\left( {a + b} \right) + c = a + \left( {b + c} \right)$.
Associative Property is true for Addition and Multiplication but not for Subtraction and Division.
Distributive Property is a property in which multiplying a sum by a number is the same as multiplying each addend with a number and then adding their products together.
Distributive Property is denoted as $\left( {a + b} \right) \times c = a \times c + b \times c$..
We can now conclude that the given equation is in the same form as the distributive form equation.
Therefore, $2 \times 13 + 8 \times 13 = \left( {2 + 8} \right) \times 13$ has the Distributive Property.
Thus, Option (D) is the correct answer.

Note: We know that Distributive Property is used to simplify the multiplication of a number by a sum or a difference. Distributive Property distributes the expression and is true for multiplication over addition and multiplication over subtraction. Here, we should keep in mind that we do not have to solve the equation, we just have to find the property it is following. So, we should compare the equation with each property.