Find two

*positive integers* X and Y, with X ≤ Y, that satisfy this set of simultaneous relationships:

lcm(X, Y)
--------- = 1785, and:
gcd(X, Y)
X + Y = 2014

well, yes, as Charlie said,
1785 = 3*5*7*17
These factors must be divided between two integers, w and z,
such that w < z and (w+z) divides 2014
There are only 8 combinations such that w < z
w z w+z 2014/(w+z)
--- ------ ----- ---------
1 3*5*7*17 1786 ~1.12
3 5*7*17 598 ~3.37
5 3*7*17 362 ~5.56
7 3*5*17 262 ~7.68
17 3*5*7 122 ~16.51
3*5 7*17 134 ~15.03
3*7 5*17 106 19 <===
5*7 3*17 86 ~23.42
Only one of these divides 2014 evenly,
so x = w*19 = 3*7*19 = **399**
and y = z*19 = 5*17*19 = **1615**