Doubly Annoying Cheryl

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them two unknowns: \(M\) and \(D.\)

\(M\) is the month of Cheryl's birthday (1-12)
\(D\) is the day of Cheryl's birthday.

Cheryl then tells Albert the sum \(M+D\) and she tells Bernard the product \(M \times D\).

Albert: "I don't know when Cheryl's birthday is, but I know that Bernard does not know too."
Bernard: "I could not figure out when Cheryl's birthday is, but I can now."
Albert: "Then I also figured out when Cheryl's birthday is."

What is the earliest Cheryl's birthday could be?

Input the month and day concatenated; e.g. September 14th is 914