Question 1204547
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I will use an idea, which is close to Alan' solution, but is slightly different.


If the sum of the reciprocals of two consecutive even integers is 23/264,

then each reciprocal is close to  {{{(1/2)*(23/264)}}} = {{{23/528}}}.


Hence, each of the two consecutive even integer is close to  {{{528/23}}} = 22.96.


From it, I conclude that the two consecutive even integer numbers are 22 and 24.


Now it is a matter of arithmetic to check that it is true.
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Solved.