How can I determine the base of the following numbers for the operations to be correct? Given:
$24_A + 17_A = 40_A$
How can I find the base of the following number $A$ so the operations are correct?

By definition of number base, we know that $24_A=2A+4$, $17_A=1A+7$, and $40_A=4A+0$, the last three terms are all in base 10. Equating them, we get $2A+4+A+7=4A$, and solving for $A$ results in $\boxed{A=11}$.
Checking by adding in base $11$, we see that $27_{11}+17_{11}=40_{11}$, because $4+7=10$ in base $11$, and carry over the $1$, $1+2+1=4$.
