23. The product of the digits of the positive integer $N$ is 20 . One of the following could not be the product of the digits of $N+1$. Which is it?
A 24
B 25
C 30
D 35
E 40
SOlUtion
D

There are two ways to make a list of digits whose product is 20 : either use $2,2,5$ and any number of $1 \mathrm{~s}$; or use 4,5 and any number of $1 \mathrm{~s}$. Either way, none of the digits of $N$ is 9 so the digits of $N+1$ will be the same as the digits of $N$ but with one of them increased by 1 . Using $2,2,5$ and any number of $1 \mathrm{~s}$ gives the following possibilities: $3,2,5$ and any number of $1 \mathrm{~s}$, with product $30 ; 2,2,6$ and any number of 1 s, with product $24 ; 2,2,5,2$ and any number of $1 \mathrm{~s}$, with product 40 . Using 4,5 and any number of 1 s gives these possibilities: 5,5 and any number of 1 s, product $25 ; 4,6$ and any number of $1 \mathrm{~s}$, product $24 ; 4,5,2$ and any number of $1 \mathrm{~s}$, product 40 . The only option given that cannot be made is 35 .