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 Simultaneous Satisfaction II (Posted on 2014-11-19)
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

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 Doing the match (spoiler) Comment 1 of 1
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
 Posted by Steve Herman on 2014-11-19 12:49:50

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