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:
- \( 19 \)
- \( 29 \)
- \( 39 \)
- \( 49 \)
- \( 59 \)
- \( 69 \)
- \( 79 \)
- \( 89 \)
- \( 99 \)
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:
\[ \boxed{59}. \]
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