The smallest whole number that is to be multiplied with 59535 to make a perfect square number is x. The sum of digits of x is

Options

A. 9
B. 5
C. 7
D. 6

Answer

To find the smallest whole number \( x \) that must be multiplied with \( 59535 \) to make it a perfect square, we will follow these steps:

### Step 1: Prime Factorization of 59535
We need to factor \( 59535 \) into its prime factors.

1. Check divisibility by \( 3 \):
- Sum of digits of \( 59535 \) is \( 5 + 9 + 5 + 3 + 5 = 27 \) (which is divisible by \( 3 \)).
- Dividing \( 59535 \) by \( 3 \):
\[ 59535 \div 3 = 19845 \]

2. Check \( 19845 \) for divisibility by \( 3 \):
- Sum of digits of \( 19845 \) is \( 1 + 9 + 8 + 4 + 5 = 27 \) (divisible by \( 3 \)).
- Dividing \( 19845 \) by \( 3 \):
\[ 19845 \div 3 = 6615 \]

3. Check \( 6615 \) for divisibility by \( 3 \):
- Sum of digits of \( 6615 \) is \( 6 + 6 + 1 + 5 = 18 \) (divisible by \( 3 \)).
- Dividing \( 6615 \) by \( 3 \):
\[ 6615 \div 3 = 2205 \]

4. Check \( 2205 \) for divisibility by \( 3 \):
- Sum of digits of \( 2205 \) is \( 2 + 2 + 0 + 5 = 9 \) (divisible by \( 3 \)).
- Dividing \( 2205 \) by \( 3 \):
\[ 2205 \div 3 = 735 \]

5. Check \( 735 \) for divisibility by \( 3 \):
- Sum of digits of \( 735 \) is \( 7 + 3 + 5 = 15 \) (divisible by \( 3 \)).
- Dividing \( 735 \) by \( 3 \):
\[ 735 \div 3 = 245 \]

6. Check \( 245 \) for divisibility by \( 5 \):
- \( 245 \) ends in \( 5 \), so it is divisible by \( 5 \).
- Dividing \( 245 \) by \( 5 \):
\[ 245 \div 5 = 49 \]

7. Factor \( 49 \):
- \( 49 = 7 \times 7 \).

So, the complete prime factorization of \( 59535 \) is:
\[ 59535 = 3^5 \times 5^1 \times 7^2 \]

### Step 2: Making it a Perfect Square
For a number to be a perfect square, all prime factors must have even exponents.

- The exponent of \( 3 \) is \( 5 \) (which is odd).
- The exponent of \( 5 \) is \( 1 \) (which is odd).
- The exponent of \( 7 \) is \( 2 \) (which is even).

To make all exponents even:
- We need one more \( 3 \) (to make \( 3^6 \)).
- We need one more \( 5 \) (to make \( 5^2 \)).

Thus, we need to multiply by:
\[ 3^1 \times 5^1 = 3 \times 5 = 15 \]

### Step 3: Finding the Sum of Digits of \( x \)
The smallest whole number \( x \) that we need to multiply with \( 59535 \) is \( 15 \).

Now, we find the sum of the digits of \( 15 \):
\[ 1 + 5 = 6 \]

### Final Answer
The sum of the digits of \( x \) is \( 6 \).

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