Buffalo Wings Puzzler solution seems more algebra-complicated than necessary?

Ok, I admit it, I have a tendencery to fall asleep during Car Talk. Not complaining, just stating the facts. I find the show relaxing I guess. So I may not have heard the solution to the “Car Talk staff eats Buffalo Wings” puzzler correctly. But it seemed like there was a lot of algebra and solving quadratic equations.

My thinking is that if the bill is $63, then the first thing I’d do is check to see if there is an integer solution. 63’s prime factors are 3 x 3 x 7.

If there is an integer solution, it seems like it has to be from among:

7 people originally, each pays $9, or 5 people, each pays $11 = $55-NO, doesn’t equal $63
9 people originally, each pays $7, or 7 people, each pays $9 = $63-YES, it does equal $63, a solution
3 people originally, each pays $21, or 1 person pays $23 = $23-NO , doesn’t equal $63
21 people originally, each pays $3, or 19 people , each pays $5 = $95-NO, doesn’t equal $63

Since only one of these combos produces the correct amount, $63, after Tom and Ray depart, isn’t the problem done at that point? The original group had 9 people is the solution. Or is the algebra – I do recall there was a missive “show your work” – required to prove that sol’n is unique?

I came up with 7 and 9 as the only reasonable answer without algebra, but it’s not much of a puzzler without having to do the math, so I understood that to be necessary. But I didn’t bother.

Good point. I guess there could be other than integer solutions, so that’s another reason for doing the math I guess.