When $-15a^{2}b{c}^3$ is divided by $-5ab{c}^2$, the quotient obtained is $3a^{2}c$.

The given statement is false.

$\frac{-15{\mathrm{a}}^2{\mathrm{bc}}^3}{-5{\mathrm{abc}}^2}$

$=\frac{-15\times \mathrm{a}\times \mathrm{a}\times \mathrm{b}\times \mathrm{c}\times \mathrm{c}\times \mathrm{c}}{-5\times \mathrm{a}\times \mathrm{b}\times \mathrm{c}\times \mathrm{c}}$

$=\frac{{\displaystyle \stackrel{3}{\cancel{-15}}\times \mathrm{a}\times \sout{\mathrm{a}\times \mathrm{b}\times \mathrm{c}\times \mathrm{c}}\times \mathrm{c}}}{\cancel{-5}\times \sout{\mathrm{a}\times \mathrm{b}\times \mathrm{c}\times \mathrm{c}}}$

$=3\times \mathrm{a}\times \mathrm{c}$

$=3\mathrm{ac}$

Hence, when $-15{\mathrm{a}}^2{\mathrm{bc}}^3$ is divided by $-5{\mathrm{abc}}^2$, the quotient obtained will be $3\mathrm{ac}$.