jmc

Since $5\times 24=120=121-1$, it follows that $-5\times 24\equiv 1(\mathrm{mod}121)$. Adding 121 to $-5$ to make it positive, we find $(-5+121)\times 24\equiv 116\times 24\equiv 1(\mathrm{mod}121)$, so it follows that the modular inverse of $24$ is $\boxed{116}$ when taken modulo $121$.