How many of the numbers from 10 through 83 have the sum of their digits equal to a perfect square? There are \( \square \) numbers whose digits sum to a perfect square.

To solve the problem, we need to find how many numbers from 10 to 83 have the sum of their digits equal to a perfect square. ### Step 1: Identify the range The numbers we are considering are from 10 to 83, inclusive. ### Step 2: Calculate the sum of digits For each number \( n \) in this range, we will calculate the sum of its digits. ### Step 3: Identify perfect squares The perfect squares less than or equal to the maximum possible sum of digits (which is \( 8 + 3 = 11 \) for the number 83) are: - \( 0^2 = 0 \) - \( 1^2 = 1 \) - \( 2^2 = 4 \) - \( 3^2 = 9 \) Thus, the perfect squares we will consider are \( 0, 1, 4, 9 \). ### Step 4: Check each number We will check each number from 10 to 83, calculate the sum of its digits, and see if it matches any of the perfect squares identified. Let's calculate the sum of digits for each number in the range and count how many of them have sums that are perfect squares. ### Step 5: Count the valid numbers We will iterate through each number from 10 to 83, calculate the sum of its digits, and check if it is a perfect square. Now, let's perform the calculations. - For \( n = 10 \): \( 1 + 0 = 1 \) (perfect square) - For \( n = 11 \): \( 1 + 1 = 2 \) - For \( n = 12 \): \( 1 + 2 = 3 \) - For \( n = 13 \): \( 1 + 3 = 4 \) (perfect square) - For \( n = 14 \): \( 1 + 4 = 5 \) - For \( n = 15 \): \( 1 + 5 = 6 \) - For \( n = 16 \): \( 1 + 6 = 7 \) - For \( n = 17 \): \( 1 + 7 = 8 \) - For \( n = 18 \): \( 1 + 8 = 9 \) (perfect square) - For \( n = 19 \): \( 1 + 9 = 10 \) - For \( n = 20 \): \( 2 + 0 = 2 \) - For \( n = 21 \): \( 2 + 1 = 3 \) - For \( n = 22 \): \( 2 + 2 = 4 \) (perfect square) - For \( n = 23 \): \( 2 + 3 = 5 \) - For \( n = 24 \): \( 2 + 4 = 6 \) - For \( n = 25 \): \( 2 + 5 = 7 \) - For \( n = 26 \): \( 2 + 6 = 8 \) - For \( n = 27 \): \( 2 + 7 = 9 \) (perfect square) - For \( n = 28 \): \( 2 + 8 = 10 \) - For \( n = 29 \): \( 2 + 9 = 11 \) - For \( n = 30 \): \( 3 + 0 = 3 \) - For \( n = 31 \): \( 3 + 1 = 4 \) (perfect square) - For \( n = 32 \): \( 3 + 2 = 5 \) - For \( n = 33 \): \( 3 + 3 = 6 \) - For \( n = 34 \): \( 3 + 4 = 7 \) - For \( n = 35 \): \( 3 + 5 = 8 \) - For \( n = 36 \): \( 3 + 6 = 9 \) (perfect square) - For \( n = 37 \): \( 3 + 7 = 10 \) - For \( n = 38 \): \( 3 + 8 = 11 \) - For \( n = 39 \): \( 3 + 9 = 12 \) - For \( n = 40 \): \( 4 + 0 = 4 \) (perfect square) - For \( n = 41 \): \( 4 + 1 = 5 \) - For \( n = 42 \): \( 4 + 2 = 6 \) - For \( n = 43 \): \( 4 + 3 = 7 \) - For \( n = 44 \): \( 4 + 4 = 8 \) - For \( n = 45 \): \( 4 + 5 = 9 \) (perfect square) - For \( n = 46 \): \( 4 + 6 = 10 \) - For \( n = 47 \): \( 4 + 7 = 11 \) - For \( n = 48 \): \( 4 + 8 = 12 \) - For \( n = 49 \): \( 4 + 9 = 13 \) - For \( n = 50 \): \( 5 + 0 = 5 \) - For \( n = 51 \): \( 5 + 1 = 6 \) - For \( n = 52 \): \( 5 + 2 = 7 \) - For \( n = 53 \): \( 5 + 3 = 8 \) - For \( n = 54 \): \( 5 + 4 = 9 \) (perfect square) - For \( n = 55 \): \( 5 + 5 = 10 \) - For \( n = 56 \): \( 5 + 6 = 11 \) - For \( n = 57 \): \( 5 + 7 = 12 \) - For \( n = 58 \): \( 5 + 8 = 13 \) - For \( n = 59 \): \( 5 + 9 = 14 \) - For \( n = 60 \): \( 6 + 0 = 6 \) - For \( n = 61 \): \( 6 + 1 = 7 \) - For \( n = 62 \): \( 6 + 2 = 8 \) - For \( n = 63 \): \( 6 + 3 = 9 \) (perfect square) - For \( n = 64 \): \( 6 + 4 = 10 \)