Sum of two positive numbers is 52 and their LCM is 168. Find the numbers?

Question

I know the answer. The numbers are 24 & 28.

I want to know how to calculate ?

Answer 1

first answer is trial and error method which is a long method

x+y= 52

x*y=168*n

where n is the hcf of 52 and 168 which is 4

two equatiions and two variables now this can be solved by substituion method

Answer 2

x + y = 52

since their LCM is 168

their product should be an integral multiple of 168

x y = 168 k ,where k is any integer

y= 168k /x

using this x + 168k/x = 52

or x^2 - 52x +168k = 0

x= { -52 +- sqrt( 52^2 - 4 * 168k ) } /2

remember k is an integer & so are x & y

thus the term inside sqrt must be perfect square

so start putting k = 1,2,3,..... & check if the term is a perfect square

here for k =4 the term is perfect square
& hence answer is 24 & 28