Question

Assume that x is an even natural number and y is a natural number such that y<x. If LCM(x,y) =2100 and GCF(x,y)=3 then find x and y

Answer 1

Investigation:
let x = 30 and y = 24
30 = 2*3*5
24 = 2*2*2*3
GCF = 6 and LCM = 120

Suppose we didn't know our original x and y, but we
know their GCF and their LCM

so we can say: x = 6a, and y = 6b
and xy = 6a(6b) = 36 ab = 120 , such that a and b are relatively prime
ab = 120/36 = 20
since a and b are relatively prime, a = 5, b = 4 , (the only choice)
and we get our original x and y with x = 6*5 = 30 and y = 4*6 = 24
.............................
now follow the same steps for yours:
we know that xy = 2100*3 = 6300
let x = 6a, and y = 6b
36ab = 6300
ab = 175
which two factor of 175 are relatively prime??
how about 25*7
so we let a = 25 ----> x = 6*25 = 150
and let b = 7 -----> y = 6*7 = 42

Answer 2

In my example, the two lines should say:

and xy = 720 , 6a(6b) = 36ab = 720 , such that a and b are relatively prime
ab = 720/36 = 20
(I typed 120 instead of 720)