Find the answer to the following questions to solve the problem

1. What will be the LCM of two numbers that are in the ratio 3 : 4?
2. What are the two numbers if the numbers are 3x and 4x and their LCM is 192?

Step 1: Find the 2 numbers using the given data

Data given

1. The ratio of the positive integers is 3 : 4
2. Their LCM is 192.
   
   

Let the two numbers be 3x and 4x.

Let us find the LCM of two numbers 3x and 4x by the prime factorization method.

Method: The LCM of the two numbers is the product of all the different prime factors of both the numbers in their respective highest powers.

Prime factorize 3x: 3 * x

Prime factorize 4x: 2^2 * x

So, the LCM of 3x and 4x = 3 * 2^2 * x = 12x

Equating 12x to the LCM of the two numbers we get, 12x = 192

Or x = 16.

So, the two numbers are 3*16 and 4*16

i.e., the numbers are 48 and 64

Step 2: Compute the difference between their squares

The numbers are 48 and 64.

The difference between their squares is 64² - 48² = (64 + 48)(64 - 48)

= 112 * 16 = 1792

  The difference between the squares of the numbers is 1792.