x' stands for a number. If sum of all the three digits of `(x!-x)` is divisible by 'x', what is 'x' ?

Options

A. 2
B. 6
C. 4
D. 12

Answer

To solve the problem, we need to find a value of \( x \) such that the sum of the digits of \( (x! - x) \) is divisible by \( x \). Let's go through the options step by step.

### Step 1: Evaluate the options

We have four options: 2, 4, 6, and 12. We will evaluate each option to see if it satisfies the condition.

### Step 2: Check \( x = 2 \)

1. Calculate \( 2! - 2 \):
\[ 2! = 2 \quad \Rightarrow \quad 2! - 2 = 2 - 2 = 0 \]
2. The sum of the digits of 0 is:
\[ 0 \]
3. Check divisibility:
\[ 0 \div 2 = 0 \quad \text{(divisible)} \]
However, since \( 0 \) is not a three-digit number, we discard this option.

### Step 3: Check \( x = 4 \)

1. Calculate \( 4! - 4 \):
\[ 4! = 24 \quad \Rightarrow \quad 4! - 4 = 24 - 4 = 20 \]
2. The sum of the digits of 20 is:
\[ 2 + 0 = 2 \]
3. Check divisibility:
\[ 2 \div 4 = 0.5 \quad \text{(not divisible)} \]
So, we discard this option.

### Step 4: Check \( x = 6 \)

1. Calculate \( 6! - 6 \):
\[ 6! = 720 \quad \Rightarrow \quad 6! - 6 = 720 - 6 = 714 \]
2. The sum of the digits of 714 is:
\[ 7 + 1 + 4 = 12 \]
3. Check divisibility:
\[ 12 \div 6 = 2 \quad \text{(divisible)} \]
This option satisfies the condition.

### Step 5: Check \( x = 12 \)

1. Calculate \( 12! - 12 \):
\[ 12! \quad \text{(a very large number, but we can simplify)} \]
We can skip the detailed calculation since we already found a valid \( x \).

Since we have already found that \( x = 6 \) satisfies the condition, we conclude that:

### Final Answer:
\[ x = 6 \]