To express the federal debt of Country C, which is $278,560,125,000, in scientific notation, we first identify the significant digits and the appropriate power of 10.
We can rewrite $278,560,125,000 as: \[ 2.78560125 \times 10^{11} \] However, for the closest estimation, we round the significant figure to one digit, which gives us: \[ 2.8 \times 10^{11} \]
Thus, the closest estimation of the federal debt of Country C, expressed in the form of a single digit times an integer power of 10, is: \[ \boxed{2.8 \times 10^{11}} \]