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
Doing the match (spoiler)