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](https://artofproblemsolving.com/wiki/index.php/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](https://artofproblemsolving.com/wiki/index.php/Maximum) of the "errors" $\| A_i-a_i\|$, the maximum [absolute value](https://artofproblemsolving.com/wiki/index.php/Absolute_value) of the difference, is as small as possible. For this minimum $M$, what is $100M$?
If any of the approximations $A_i$ is less than 2 or more than 3, the error associated with that term will be larger than 1, so the largest error will be larger than 1.  However, if all of the $A_i$ are 2 or 3, the largest error will be less than 1.  So in the best case, we write 19 as a sum of 7 numbers, each of which is 2 or 3.  Then there must be five 3s and two 2s.  It is clear that in the best appoximation, the two 2s will be used to approximate the two smallest of the $a_i$, so our approximations are $A_1 = A_2 = 2$ and $A_3 = A_4 = A_5 = A_6 = A_7 = 3$ and the largest error is $|A_2 - a_2| = 0.61$, so the answer is $\boxed{061}$.