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

lcm(X, Y) + gcd(X, Y) = 53314, and:

X + Y = 2014

Well, the gcd(X,Y) divides X and Y and lcm(X,Y), so it must also divide 2014 and 53314.

2014 = 2*19*53

53314 = 2*19*23*61

So the gcd(X,Y) must be 1 or 2 or 19 or 38.

But gcd*lcm = X*Y

so

(53314-gcd)*gcd = X*(2014-X)

X^2 -2014X + (53314-gcd)*gcd = 0

X = .5*(2014 +/- sqrt(2014*2014 - 4*(53314-gcd)*gcd))

Let's try the 4 possible values of gcd in

sqrt(2014*2014 - 4*(53314-gcd)*gcd)).

Only gcd = 19 is rational, giving a sqrt of 76,

so X = (2014 +/- 76)/2 = **969 and 1045**

These must be X and Y

Checking:

gcd(969,1045) = 19

lcm(969,1045) = 969*1045/19 = 53295

53295 + 19 = 53314

969 + 1045 = 2014

Of course, I fat-fingered the subject. I meant "doing the math"

*Edited on ***November 19, 2014, 1:13 pm**