Question 912032
<pre>
{{{1/x+1/(x+1)+1/(x+2)=37/60}}}

Multiply through by LCD = 60x(x+1)(x+2)

{{{1*60x(x+1)(x+2)/x+1*60x(x+1)(x+2)/(x+1)+1*60x(x+1)(x+2)/(x+2)=37*60x(x+1)(x+2)/60}}}

{{{60(x+1)(x+2)+60x(x+2)+60x(x+1)=37x(x+1)(x+2)}}}

{{{60(x^2+3x+2)+60x^2+120x+60x^2+60x=37x(x^2+3x+2)}}}

{{{60x^2+180x+120+60x^2+120x+60x^2+60x=37x^3+111x^2+74x}}}

{{{180x^2+360x+120=37x^3+111x^2+74x}}}

{{{0=37x^3-69x^2-286x-120}}}

Feasible rational solutions are the fractions with only 1
as a denominator (integers) since 37 is not a factor of 
any factor of 120.

Factors of 120 are 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

One solution is 4

4|37  -69 -286 -120
 |<u>    148  316  120</u>
  37   79   30    0

{{{(x-4)(37x^2+79x+30)=0}}}

The quadratic does not factor, thus any other
solution would not be an integer.

Answer: 4,5,6

Edwin</pre>