Question

Find the value of the following: $81265\times 169-82165\times 69$ .

Answer 1

Hint: The given problem is related to properties of operations of integers. 

Integers have five properties of operation, which are:
Closure Property: The closure property states that the product of two integers will always be an integer.  
Associative property: The associative property states that the multiplication of three integers can be done in any order, i.e. $x\times \left( y\times z \right)=\left( x\times y \right)\times z$ .
Identity property: Identity property states that when an integer is multiplied by 1, the product is the same as the initial integer, i.e. $1\times x=x$. 
Commutative property: The commutative property states that the position of integers with respect to the multiplication sign does not affect the product, i.e. $x\times y=y\times x$ . 
Distributive property: The distributive property states that multiplication can be distributed over another mathematical operation within a bracket, i.e. $x\left( y+z \right)=xy+xz$ .

We will check the given expression, and apply the required property to get the answer.

Complete step by step answer:

We are given $81265\times 169-82165\times 69$ .

To evaluate the given expression, we use the distributive property of multiplication, which states that multiplication can be distributed over another mathematical operation within a bracket, $x\left( y+z \right)=xy+xz$ which is same as $xy+xz= x(y+z)$. 

The given expression using distributive property can be written as

$=81625\times \left( 169-69 \right)$

On simplification,  
$=81625\times \left( 100 \right)$
$=8162500$
Hence, the value of the given expression $81265\times 169-82165\times 69$ is $8162500.$

Note: To solve this type of question, first we need to check which property can be applied to get the answer quickly. While using distributive property, do not change the sign. It will lead to wrong answers. Also, calculations should be done carefully. Distributive property helps in fast calculation of products and sum of two numbers.