Total number of natural numbers being the perfect square whose root is equal to the sum of the digits of the perfect square is :

Options

A. a. 0
B. b. 1
C. c. 2
D. d. 12

Answer

To solve the problem, we need to find the total number of natural numbers that are perfect squares, where the square root of the perfect square is equal to the sum of its digits. Let's go through the steps to find these numbers.

### Step 1: Understand the Problem
We need to identify perfect squares \( n^2 \) such that the sum of the digits of \( n^2 \) equals \( n \).

### Step 2: Identify Perfect Squares
The first few perfect squares are:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
- \( 9^2 = 81 \)
- \( 10^2 = 100 \)
- and so on...

### Step 3: Calculate the Sum of Digits
We will calculate the sum of the digits for each perfect square and check if it equals the square root.

1. For \( 1^2 = 1 \):
- Sum of digits = \( 1 \) (which equals \( 1 \))

2. For \( 2^2 = 4 \):
- Sum of digits = \( 4 \) (which does not equal \( 2 \))

3. For \( 3^2 = 9 \):
- Sum of digits = \( 9 \) (which does not equal \( 3 \))

4. For \( 4^2 = 16 \):
- Sum of digits = \( 1 + 6 = 7 \) (which does not equal \( 4 \))

5. For \( 5^2 = 25 \):
- Sum of digits = \( 2 + 5 = 7 \) (which does not equal \( 5 \))

6. For \( 6^2 = 36 \):
- Sum of digits = \( 3 + 6 = 9 \) (which does not equal \( 6 \))

7. For \( 7^2 = 49 \):
- Sum of digits = \( 4 + 9 = 13 \) (which does not equal \( 7 \))

8. For \( 8^2 = 64 \):
- Sum of digits = \( 6 + 4 = 10 \) (which does not equal \( 8 \))

9. For \( 9^2 = 81 \):
- Sum of digits = \( 8 + 1 = 9 \) (which equals \( 9 \))

### Step 4: Compile Results
From our calculations, the perfect squares that meet the criteria are:
- \( 1 \) (from \( 1^2 \))
- \( 81 \) (from \( 9^2 \))

### Step 5: Count the Valid Numbers
The total count of such natural numbers is \( 2 \) (which are \( 1 \) and \( 81 \)).

### Final Answer
The total number of natural numbers being the perfect square whose root is equal to the sum of the digits of the perfect square is **2**.

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