SOLUTION: Show that the sum of the reciprocals of three different positive integers is greater than 6 times the reciprocal of the product.

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Question 1137859: Show that the sum of the reciprocals of three different positive integers is greater than 6 times the reciprocal of the product.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

If a,b,c are the 3+positive integers
1%2Fa+%2B1%2Fb+%2B1%2Fc+%3E+6%2Fabc
%28bc%2Bac%2Bab%29%2Fabc+%3E6%2Fabc+
so
%28bc%2Bac%2Bab%29%3E6

The lowest positive integers that are different are 1,2,3.

so, the lowest value that %28bc%2Bac%2Bab%29 could have is 1%2A2%2B2%2A3%2B1%2A3=2%2B6%2B3=+11 and 11%3E6
therefore
1%2Fa+%2B1%2Fb+%2B1%2Fc+%3E+6%2Fabc is true+