Given that Alpha's number is 4, 5, 8 or 10, and Beta's number is 4, 6, 7 or 9, there are 16 possibilities, of which only four allow Gamma to deduce Alpha's number, namely when Gamma's number is 5, 6, 9 or 10. The two cases where Gamma's number is 5 or 10 are interesting, because they both allow Gamma to know that Alpha's number is 4, but Alpha doesn't have enough information to reduce the shortlist to just one case. As soon as Alpha reveals that situation, Beta can deduce what Alpha's number is, but we don't need to know Beta's exclamation about that.

We already know that Alpha's number is 4. Beta's number may be 9 (in which case, Gamma's number is 5) or 6 (in which case, Gamma's number is 10).