SOLUTION: find three consecutive integers the sum of whose reciprocal is 37/60

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Question 912032: find three consecutive integers the sum of whose reciprocal is 37/60

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
1%2Fx%2B1%2F%28x%2B1%29%2B1%2F%28x%2B2%29=37%2F60

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



60%28x%2B1%29%28x%2B2%29%2B60x%28x%2B2%29%2B60x%28x%2B1%29=37x%28x%2B1%29%28x%2B2%29

60%28x%5E2%2B3x%2B2%29%2B60x%5E2%2B120x%2B60x%5E2%2B60x=37x%28x%5E2%2B3x%2B2%29

60x%5E2%2B180x%2B120%2B60x%5E2%2B120x%2B60x%5E2%2B60x=37x%5E3%2B111x%5E2%2B74x

180x%5E2%2B360x%2B120=37x%5E3%2B111x%5E2%2B74x

0=37x%5E3-69x%5E2-286x-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
 |    148  316  120
  37   79   30    0

%28x-4%29%2837x%5E2%2B79x%2B30%29=0

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

Answer: 4,5,6

Edwin