Use the rank-nullity theorem. It says that the rank of a matrix [tex]\mathbf A[/tex], [tex]\mathrm{rank}(\mathbf A)[/tex], has the following relationship with its nullity [tex]\mathrm{null}(\mathbf A)[/tex] and its number of columns [tex]n[/tex]:
[tex]\mathrm{rank}(\mathbf A)+\mathrm{null}(\mathbf A)=n[/tex]
We're given that [tex]\mathbf A[/tex] is [tex]13\times91[/tex], i.e. has [tex]n=91[/tex] columns. The largest rank that a [tex]m\times n[/tex] matrix can have is [tex]\min\{m,n\}[/tex]; in this case, that would be 13.
So if we take [tex]\mathbf A[/tex] to be of rank 13, i.e. we maximize its rank, we must simultaneously be minimizing its nullity, so that the smallest possible value for [tex]\mathrm{null}(\mathbf A)[/tex] is given by
[tex]13+\mathrm{null}(\mathbf A)=91\implies\mathrm{null}(\mathbf A)=91-13=78[/tex]
