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 Simultaneous Satisfaction (Posted on 2014-09-27)
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```

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution and outline of a possible analytic method | Comment 1 of 2
efDbl A-Z
Dim crlf\$

ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
Text1.Text = ""
crlf\$ = Chr(13) + Chr(10)
Form1.Visible = True
DoEvents

Sum = 2014
For x = 1 To Sum / 2
y = Sum - x
l = lcm(x, y)
g = gcd(x, y)
If l / g = 1785 Then
Text1.Text = Text1.Text & x & Str(y) & "     " & lcm(x, y) & Str(gcd(x, y)) & crlf
End If
Next

Text1.Text = Text1.Text & crlf & " done"
End Sub

Function gcd(a, b)
x = a: y = b
Do
q = Int(x / y)
z = x - q * y
x = y: y = z
Loop Until z = 0
gcd = x
End Function

Function lcm(a, b)
lcm = a * b / gcd(a, b)
End Function

finds

x=399 y=1615     LCM=33915 GCD=19

LCM/GCD = 1785 as required

LCM/GCD is of course also x * y / GCD ^ 2

1785 = 3 * 5 * 7 * 17

and

x = 3 * 7 * 19    and y = 5 * 17 * 19

so an analytic method might have employed splitting the factors of 1785 into the two separate x and y and adding a common factor identically to each, so that the sum of 2014 would be reached.

 Posted by Charlie on 2014-09-27 15:17:56

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