Let x be the least numbers, which when divided by 5, 6, 7 and 8 leaves a remainder 3 in each case but when divided by 9 leaves no remainder. The sum of digits of x is

Options

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

Answer

To solve the problem step by step, we need to find the least number \( x \) that meets the given conditions.

### Step 1: Understand the conditions
The number \( x \) must satisfy the following:
- When divided by 5, 6, 7, and 8, it leaves a remainder of 3.
- When divided by 9, it leaves no remainder.

### Step 2: Set up the equation
Since \( x \) leaves a remainder of 3 when divided by 5, 6, 7, and 8, we can express \( x \) in the form:
\[ x = LCM(5, 6, 7, 8)k + 3 \]
where \( k \) is a positive integer.

### Step 3: Calculate the LCM
To find the least common multiple (LCM) of 5, 6, 7, and 8:
- The prime factorization of each number:
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)
- \( 8 = 2^3 \)

The LCM takes the highest power of each prime:
- \( 2^3 \) from 8
- \( 3^1 \) from 6
- \( 5^1 \) from 5
- \( 7^1 \) from 7

Thus,
\[ LCM(5, 6, 7, 8) = 2^3 \times 3^1 \times 5^1 \times 7^1 = 840 \]

### Step 4: Write the expression for \( x \)
Now we can express \( x \):
\[ x = 840k + 3 \]

### Step 5: Check divisibility by 9
We need \( x \) to be divisible by 9:
\[ 840k + 3 \equiv 0 \ (\text{mod} \ 9) \]
Calculating \( 840 \mod 9 \):
\[ 840 \div 9 = 93 \quad \text{(remainder 3)} \]
Thus,
\[ 840 \equiv 3 \ (\text{mod} \ 9) \]
So,
\[ 840k + 3 \equiv 3k + 3 \equiv 0 \ (\text{mod} \ 9) \]
This simplifies to:
\[ 3(k + 1) \equiv 0 \ (\text{mod} \ 9) \]
This means \( k + 1 \) must be divisible by 3.

### Step 6: Find suitable values for \( k \)
Let’s try \( k = 2 \) (the smallest value that satisfies \( k + 1 \equiv 0 \ (\text{mod} \ 3) \)):
\[ x = 840 \times 2 + 3 = 1680 + 3 = 1683 \]

### Step 7: Verify divisibility by 9
Now, check if 1683 is divisible by 9:
Sum of the digits of 1683:
\[ 1 + 6 + 8 + 3 = 18 \]
Since 18 is divisible by 9, \( x = 1683 \) meets all conditions.

### Step 8: Find the sum of the digits of \( x \)
The sum of the digits of \( x = 1683 \) is:
\[ 1 + 6 + 8 + 3 = 18 \]

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

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