Let x be the least number, which when divided by 5, 6, 7 and 8 leaves a remainder 3 in each case but when divided by 9 leaves remainder 0. The sum of digits of x is:

Options

A. 24
B. 21
C. 22
D. 18

Answer

To solve the problem, we need to find the least number \( x \) that satisfies the following conditions:

1. When divided by 5, 6, 7, and 8, it leaves a remainder of 3.
2. When divided by 9, it leaves a remainder of 0.

### Step-by-Step Solution:

**Step 1: Understand the conditions.**

We know that:
- \( x \equiv 3 \mod 5 \)
- \( x \equiv 3 \mod 6 \)
- \( x \equiv 3 \mod 7 \)
- \( x \equiv 3 \mod 8 \)
- \( x \equiv 0 \mod 9 \)

This means that \( x - 3 \) must be divisible by 5, 6, 7, and 8.

**Step 2: Find the LCM of 5, 6, 7, and 8.**

To find \( x - 3 \), we first need to calculate the least common multiple (LCM) of 5, 6, 7, and 8.

- The prime factorization is:
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)
- \( 8 = 2^3 \)

Taking the highest powers of all prime factors:
- \( 2^3 \) from 8,
- \( 3^1 \) from 6,
- \( 5^1 \) from 5,
- \( 7^1 \) from 7.

Thus, the LCM is:
\[ \text{LCM}(5, 6, 7, 8) = 2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 \]
Calculating this step-by-step:
- \( 8 \times 3 = 24 \)
- \( 24 \times 5 = 120 \)
- \( 120 \times 7 = 840 \)

So, \( \text{LCM}(5, 6, 7, 8) = 840 \).

**Step 3: Express \( x \) in terms of the LCM.**

Since \( x - 3 \) is a multiple of 840, we can write:
\[ x - 3 = 840k \quad \text{for some integer } k \]
Thus,
\[ x = 840k + 3 \]

**Step 4: Use the second condition.**

Now, we need \( x \equiv 0 \mod 9 \):
\[ 840k + 3 \equiv 0 \mod 9 \]
Calculating \( 840 \mod 9 \):
- \( 840 = 8 + 4 + 0 = 12 \)
- \( 12 \mod 9 = 3 \)

So,
\[ 3k + 3 \equiv 0 \mod 9 \]
This simplifies to:
\[ 3(k + 1) \equiv 0 \mod 9 \]
Thus, \( k + 1 \) must be a multiple of 3:
\[ k + 1 = 3m \quad \text{for some integer } m \]
This gives:
\[ k = 3m - 1 \]

**Step 5: Substitute back to find \( x \).**

Substituting \( k \) back into the equation for \( x \):
\[ x = 840(3m - 1) + 3 = 2520m - 840 + 3 = 2520m - 837 \]

**Step 6: Find the least value of \( x \).**

To find the least positive \( x \), we set \( m = 1 \):
\[ x = 2520(1) - 837 = 2520 - 837 = 1683 \]

**Step 7: Calculate the sum of the digits of \( x \).**

Now, we find the sum of the digits of 1683:
\[ 1 + 6 + 8 + 3 = 18 \]

### Final Answer:
The sum of the digits of \( x \) is \( \boxed{18} \).