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## SAT Number Properties Question: LCM Ratio & Squares

### Question 3: Numbers : LCM and Ratios

Two positive integers are in the ratio 3 : 4. If their LCM is 192, what is the difference between the squares of the numbers?

1. 1792
2. 1790
3. 1798
4. 1796
5. 1794
Choice A. The difference between the squares of the numbers is 1792.

This one is a number properties question and tests your understanding of Least Common Multiple of two numbers. Clues / hints to crack the question are given in the first tab. Using those try and solve the question yourself. If you are not able to solve using the hint, click on the next tabs to get a detailed explanation to this number properties question.

### 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.