How many 3 digit numbers have the property that the middle digit is the product of the first and last digits?

Suppose the number begins with $3$. Then it must end with a digit small enough so that the product (which would be the middle digit) is $<10$. That admits $1+[9/3]=4$ choices where $[x]$ is the greatest integer less than or equal to $x$ and the possibility of a zero units digit is included. Do this for all nine possible initial digits and you get

$9+[9/1]+[9/2]+[9/3]+...+[9/9]=32$

Possible 'end digit' pairs: $$(1,0), (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9),\\ (2,0), (2,1), (2,2), (2,3), (2,4),\\ (3,0), (3,1), (3,2), (3,3),\\ (4,0), (4,1), (4,2),\\ (5,0), (5,1),\\ (6,0), (6,1),\\ (7,0), (7,1),\\ (8,0), (8,1),\\ (9,0), (9,1)$$ so I get $32$.

I'm assuming that a '$3$-digit number' cannot have a $0$ as its first digit, otherwise you get another ten options $(0,i)$.