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.

Question

UpStudy AI · Accepted Answer

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 $$ n = 64 $$: $$ 6 + 4 = 10 $$

There are 12 numbers between 10 and 83 where the sum of their digits is a perfect square.

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.

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