Guess the Sum

Note first that since the sum can be from $3$ to $17$ we are dealing with non-zero digits.

For Betty to be sure of what the sum of the two digits is, based on the knowledge of the last digit of the product of the two digits, this last digit must be unique over all products of two digits. Now

• $2$ is the last digit in the products $3 \times 4$ and $2 \times 6$ , among others,

• $3$ is the last for $1 \times 3$ and $7 \times 9$ ,

• $4$ is the last for $1 \times 2$ and $2 \times 7$ ,

• $5$ is the last for $3 \times 5$ and $5 \times 7$ ,

• $6$ is the last for $1 \times 6$ and $2 \times 3$ ,

• $7$ is the last for $1 \times 7$ and $3 \times 9$ ,

• $8$ is the last for $2 \times 4$ and $6 \times 8$ .

However, $1$ is the last digit for just one product, namely $3 \times 7$ , and $9$ is the last digit for only one product, namely $1 \times 9$ . In either case the sum of the digits is $10$ , so whether the last digit of Betty's phone number is $1$ or $9$ , she will be ale to conclude that the sum of Alan's two distinct digits is $\boxed{10}$ .