Are all numbers good?

A two-digit positive integer, whose sum of digits and product of digits add up to the number itself is called a good number. Find the number of good numbers.

Details and assumptions:

  • I am asking for only the first sum of digits, not the digital sum, which is an iterated process.
  • Both digits need not be distinct, but a leading zero is not allowed.

Difficulty: Small, but pointy.  \boxed{\text{Difficulty: }\color{#D61F06}{\text{Small, but pointy.}} \quad \Large \color{#D61F06}{\dagger~ \dagger~ }\color{grey}{\dagger\dagger \dagger}}

This problem is completely original.

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