By what number should $(-5)^{-1}$ be divided to give the quotient $(-25)^{-1}$.

Let the number be $x$. So,

$(-5)^{-1} ÷ x = (-25)^{-1}\\[1em] \Rightarrow \dfrac{-1}{5} ÷ x = \dfrac{-1}{25}\\[1em] \Rightarrow \dfrac{-1}{5} \times \dfrac{1}{x} = \dfrac{-1}{25}\\[1em] \Rightarrow -\dfrac{1}{5x} = -\dfrac{1}{25}\\[1em] \Rightarrow -5x = 25 \\[1em] \Rightarrow x = \dfrac{25}{5} \\[1em] \Rightarrow x = 5$

Hence, (-5)^-1 should be divided by 5 to give the quotient as (-25)^-1.