Hide answer & extension
Any number can be written as \(10a+27b\) with integer \(a\) and \(b\), since \(1=3\times27-8\times10\).
So the problem may be thought of as asking when one of \(a\) and \(b\) must be negative.
Given one way of writing a number, you can get the others by shifting by 14*29. For example,
$$219 = 10\times30 - 27\times3$$ $$= 10 + 10\times29 - 27\times3 $$ $$= 1\times10 + 27\times11$$
So the question now becomes: When does this adjustment fail to eliminate negative numbers?
This is when you are at what Pedro calls "limit coefficients":
$$10\times(-1) + 27\times13 = 10\times28 + 27\times(-1) = 233$$
So the answer is 233.
Extension
Let \(n\) and \(m\) be integers. What is the largest number that cannot be written in the form \(na+mb\), where \(a\) and \(b\) are nonnegative integers?