Question

The sum of two numbers is 684 and their H.C.F is 57. The number of possible pairs of such numbers is

• A. $2$
• B. $3$
• C. $4$
• D. None

Answer 1

Answer: (A)

$2$

Let the two numbers be $x$ and $y$ respectively.

It is given that the sum of the two numbers is $684$, therefore,

$x+y=684$

Also $57$ is their HCF, thus both numbers must be divisible by $57$.

So, let $x=57a$ and $y=57b$, then

$57a+57b=684\Rightarrow 57(a+b)=684\Rightarrow a+b=\frac{684}{57}\Rightarrow a+b=12$

Therefore, required possible pair of values of $x$ and $y$ which are prime to each other are $(1,11)$ and $(5,7)$.

Thus, the required numbers are $(57,627)$ and $(285,399)$.

Hence, the number of possible pairs is $2$.