Defining the DT LTI system
$ x[n] \rightarrow system \rightarrow y[n] = 5x[n] $
a) Finding the unit impulse response h[n] and the system function F(z).
$ x[n] = \delta [n] \rightarrow system \rightarrow y[n]=5\delta [n] $
Therefore the unit impulse response, $ h[n] = 5\delta [n] $
For a DT LTI system,
$ Z^n \rightarrow system \rightarrow F(z)Z^n $
Output of the system, $ F(z)Z^n = h[n]*Z^n = \sum_{m=-\infty}^{\infty} h[m]Z^{n-m} = Z^n\sum_{-\infty}^{\infty}h[m]Z^{-m} $
Therefore, $ F(z) = \sum_{-\infty}^{\infty}h[m]Z^{-m} = \sum_{-\infty}^{\infty}5\delta [m] Z^{-m} $
