Question

What is the arithmetic mean of all of the positive two-digit integers with the property that the integer is equal to the sum of its first digit plus its second digit plus the product of its two digits?

Answer 1

To find the arithmetic mean of all the positive two-digit integers \( ab \) (where \( a \) is the first digit and \( b \) is the second digit), we need to satisfy the property:

\[
10a + b = a + b + ab.
\]

This simplifies to:

\[
10a + b = a + b + ab \implies 10a + b - a - b = ab \implies 9a = ab \implies ab - 9a = 0 \implies a(b - 9) = 0.
\]

Since \( a \) (the first digit of a two-digit number) cannot be \( 0 \), we must have:

\[
b - 9 = 0 \implies b = 9.
\]

This means that any two-digit integer of the form \( ab \) must have \( b = 9 \). Thus, the possible integers are:

- \( 19 \)
- \( 29 \)
- \( 39 \)
- \( 49 \)
- \( 59 \)
- \( 69 \)
- \( 79 \)
- \( 89 \)
- \( 99 \)

Now we can enumerate these integers:

1. \( 19 \)
2. \( 29 \)
3. \( 39 \)
4. \( 49 \)
5. \( 59 \)
6. \( 69 \)
7. \( 79 \)
8. \( 89 \)
9. \( 99 \)

Next, we need to calculate the arithmetic mean. The sum of these integers:

\[
19 + 29 + 39 + 49 + 59 + 69 + 79 + 89 + 99.
\]

Calculating this sum step-by-step:

\[
19 + 29 = 48, \\
48 + 39 = 87, \\
87 + 49 = 136, \\
136 + 59 = 195, \\
195 + 69 = 264, \\
264 + 79 = 343, \\
343 + 89 = 432, \\
432 + 99 = 531.
\]

Now, we find the number of elements in our set, which is \( 9 \). Therefore, the arithmetic mean is:

\[
\text{Arithmetic Mean} = \frac{\text{Sum}}{\text{Count}} = \frac{531}{9}.
\]

To simplify \( \frac{531}{9} \), we perform the division:

\[
531 \div 9 = 59 \quad \text{remainder } 0.
\]

Thus, \( 531 \) is evenly divisible by \( 9 \), and:

\[
\frac{531}{9} = 59.
\]

Consequently, the arithmetic mean of all of the positive two-digit integers with the property described is:

\[
\boxed{59}.
\]