The HCF and the LCM of Two numbers are 19 and 342, respectively. If one of the numbers is $$\frac{9}{2}$$ times of the other, What is the greatest number.

Correct option is D

Given:

The HCF (Highest Common Factor) of two numbers is 19.

The LCM (Least Common Multiple) of the two numbers is 342.

One of the numbers is $$\frac{9}{2}$$​ times the other.

Concept Used:

For two numbers, the product of their HCF and LCM is equal to the product of the numbers themselves 

LCM × HCF = product of the numbers 

Solution:

Let the two numbers be x and y, and assume that x = $$\frac{9}{2}$$y

Substituting the given values for the HCF, LCM and x  :

19 × 342 = $$\frac{9}{2}y$$​ × y 

​$$19 \times 342 = \frac{9}{2} y^2$$ 

$$6498 = \frac{9}{2} y^2$$ 

​$$y^2 = \frac{(6498×2)}{9}$$ 

​$$y^2 = 1444$$ 

Take the square root of both sides:

​$$y = \sqrt{1444} = 38$$ 

Since $$x = \frac{9y}{2}$$, we can find x by substituting y = 38: 

​$$x = \frac{9}{2} \times 38 = 171$$ 

The two numbers are 171 and 38. The greatest number is 171.

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