The sum of three consecutive integers is 5685. Which of the following is the correct set of these numbers? (a) 1893, 1894, 1895 (b) 1895, 1896, 1897 (c) 1899, 1900, 1901 (d) 1897, 1898, 1899 (e) None of these

Given the sum of three consecutive integers is 5685, we express this as:

\[
n + (n + 1) + (n + 2) = 5685
\]

Simplifying, we get:

\[
3n + 3 = 5685
\]
\[
3n = 5682
\]
\[
n = \frac{5682}{3} = 1894
\]

However, the value of \(n\) calculated here represents the smallest of the three consecutive numbers, not the middle one. This correction leads to the correct identification of the series as follows:

• Smallest number: \(1894\)
• Middle number: \(1894 + 1 = 1895\)
• Largest number: \(1894 + 2 = 1896\)

Therefore, the correct set of numbers is \(1894, 1895, 1896\), which does not match any of the options provided explicitly as listed. Given the specific numbers and the correction in understanding that \(n\) represents the starting (smallest) number in the sequence, the correct response should accurately reflect this set.

Thus, the corrected and accurate answer is (e) None of these, as the exact sequence derived from the calculation does not appear in the options provided.