The sum and difference of LCM and HCF of 2 numbers is 638 and 580. The sum of two numbers is 290. What are the two numbers?

Let the L.C.M. and H.C.F. of the two numbers be $$x$$ and $$y$$

According to ques, => $$x+y=638$$

and $$x-y=580$$

Adding above equations, we get : $$x=609$$ and $$y=29$$

Thus, H.C.F. = 29 and L.C.M. = 609

Let the two numbers be $$29a$$ and $$29b$$, where $$a$$ and $$b$$ are co-prime

=> Sum of numbers = $$29a+29b=290$$

=> $$a+b=10$$

Since, they are co-prime, possible values of $$(a,b)$$ are : (1,9), (3,7)

Now, to get L.C.M. as 609, we have $$a=3$$ and $$b=7$$ (or vice-versa)

Thus, the numbers are : 87 and 203

=> Ans - (B)