Let x denote the greatest 4-digit number which when divided by 6, 7, 8, 9 and 10 leaves a remainder of 4, 5, 6, 7 and 8 respectively. Then, the sum of the four-digits of is X. Then, the sum of the four-digits of x is

Options

A. 25
B. 18
C. 20
D. 22

Answer

To solve the problem, we need to find the greatest 4-digit number \( x \) that, when divided by 6, 7, 8, 9, and 10, leaves remainders of 4, 5, 6, 7, and 8, respectively.

### Step-by-Step Solution:

1. **Understanding the Remainders**:
The problem states that:
- \( x \mod 6 = 4 \)
- \( x \mod 7 = 5 \)
- \( x \mod 8 = 6 \)
- \( x \mod 9 = 7 \)
- \( x \mod 10 = 8 \)

This can be rewritten as:
- \( x + 2 \equiv 0 \mod 6 \)
- \( x + 2 \equiv 0 \mod 7 \)
- \( x + 2 \equiv 0 \mod 8 \)
- \( x + 2 \equiv 0 \mod 9 \)
- \( x + 2 \equiv 0 \mod 10 \)

This means that \( x + 2 \) must be a common multiple of 6, 7, 8, 9, and 10.

2. **Finding the LCM**:
We need to find the least common multiple (LCM) of these numbers:
- The prime factorization is:
- \( 6 = 2 \times 3 \)
- \( 7 = 7 \)
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 10 = 2 \times 5 \)

The LCM is obtained by taking the highest power of each prime:
- \( LCM = 2^3 \times 3^2 \times 5 \times 7 = 2520 \)

3. **Finding the Greatest 4-Digit Number**:
The greatest 4-digit number is 9999. We need to find the largest multiple of 2520 that is less than or equal to 9999.
- Dividing 9999 by 2520 gives approximately \( 3.968 \).
- The largest integer \( k \) is 3, so we calculate:
\[ 2520 \times 3 = 7560 \]

4. **Adjusting for the Remainders**:
Now, we need to adjust \( 7560 \) to find \( x \):
\[ x = 7560 - 2 = 7558 \]

5. **Verifying the Remainders**:
We check if \( 7558 \) gives the correct remainders:
- \( 7558 \mod 6 = 4 \)
- \( 7558 \mod 7 = 5 \)
- \( 7558 \mod 8 = 6 \)
- \( 7558 \mod 9 = 7 \)
- \( 7558 \mod 10 = 8 \)

All conditions are satisfied.

6. **Calculating the Sum of the Digits**:
Now, we find the sum of the digits of \( 7558 \):
\[ 7 + 5 + 5 + 8 = 25 \]

### Final Answer:
The sum of the four digits of \( x \) is \( 25 \).