Write 100 as the sum of two positive integers 
Write $100$ as the sum of two positive integers, one of them being a multiple of $7$, while the other is a multiple of $11$.

Since $100$ is not a big number, I followed the straightforward reasoning of writing all multiples up to $100$ of either $11$ or $7$, and then finding the complement that is also a multiple of the other. So then 
$100 = 44 + 56 = 4 \times 11 + 8 \times 7$. 
But is it the smart way of doing it? Is it the way I was supposed to solve it? I'm thinking here about a situation with a really large number that turns my plug-in method sort of unwise.

While certainly not the ideal solution, this problem is certainly in the realm of Integer Programming. As plenty of others have pointed out, there are more direct approaches. However, I suspect ILP solvers would operate quite efficiently in your case, and requires less 'thought capital'.
