A number $N$ in base 10 is $503$ in base $b$ and $305$ in base $b+2$. What is the product of the digits of $N$?

We write the base expansions for $N$:

1. $N=5b^2+0b+3=5b^2+3$

2. $N=3(b+2{)}^2+0(b+2)+5=3(b^2+4b+4)+5=3b^2+12b+17$

Equating the two expressions for $N$:

Divide by 2:

Since the base $b$ must be a positive integer, we have $b=7$.

Substitute $b=7$ back into the first equation to find $N$:

The digits of $N=248$ are $2,4,$ and $8$.
The product of the digits is: