Question

The sum of the squares of the LCM and the GCD of two natural numbers is 2308. If the product of the two natural numbers is 96, what could the sum of their LCM and GCD be?

• A. 32
• B. 25
• C. 49
• D. 50

Answer 1

The correct option is D 50

Let the two natural numbers be "a" and "b".

Also, we know that:

$LC{M}^2+GC{D}^2=2308$

Now, calculating the square of the sum of the LCM and the GCD, we get:

$(LCM+GCD{)}^2=LC{M}^2+GC{D}^2+2\times (LCM\times GCD)=2308+2\times (LCM\times GCD)$

Now, the product of the LCM and the GCD of two numbers will always be the same as the product of the numbers themselves.

$⟹LCM\times GCD=a\times b$

But it is given that the product of the two numbers is 96.

$⟹a\times b=96$

$⟹(LCM+GCD{)}^2=2308+2\times 96=2308+192=2500$

$⟹(LCM+GCD)=\sqrt{2500}=50$