Elementary problems that would've been hard for past mathematicians, but are easy to solve today? I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with. 
Some examples that come to mind are group testing problems, which would be difficult to solve without a notion of error-correcting codes, and -- for even earlier mathematicians -- calculus questions such as calculating the area  of some $n$-dimensional body.
The questions have to be understandable to older mathematicians and elementary in some sense. That is, past mathematicians should be able to appreciate them just as well as we can. 

I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level.
Ph. Davis and R. Hersh tell the long and painful story of  Fourier series. 
I quote from their book:

"Fourier didn't know Euler had already done this, so he did it over. And Fourier, like Bernoulli and Euler before him,  overlooked the beautifully direct method of orthogonality [...]. Instead, he went through an incredible computation, that could serve as a classic  example of physical insight leading to the right answer in spite of flagrantly wrong reasoning." 

(Fifth Ch. "Fourier Analysis".)
