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?
I want to find all the positive two-digit integers that satisfy the given property, and then average them. Let's call the integer x, and let its first digit be a and its second digit be b. Then x = 10a + b, and the property says that x = a + b + ab. So I need to solve the equation 10a + b = a + b + ab for a and b. I can rearrange this equation to get 9a = ab, and then divide both sides by a to get 9 = b. This means that b must be 9, and a can be any digit from 1 to 9. So the possible values of x are 19, 29, 39, 49, 59, 69, 79, 89, and 99. To find the mean of these numbers, I can add them up and divide by the number of terms. The sum is 19 + 29 + 39 + 49 + 59 + 69 + 79 + 89 + 99 = 531. The number of terms is 9. So the mean is 531/9 = 59. 