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

Posted by Steve Herman on 2014-09-27 20:29:23