1985 AIME Problem 8

The sum of the following seven numbers is exactly 19: a_1 = 2.56, a_2 = 2.61, a_3 = 2.65, a_4 = 2.71, a_5 = 2.79, a_6 = 2.82, a_7 = 2.86. It is desired to replace each a_i by an integer approximation A_i, 1\le i \le 7, so that the sum of the A_i's is also 19 and so that M, the maximum of the "errors" \lvert A_i-a_i \rvert, is as small as possible. For this minimum M, what is 100M?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.