What is the least value of K so the so that the number 6735k1is divisible by 9?

Options

A. 5
B. 7
C. 4
D. 3

Answer

To find the least value of \( K \) such that the number \( 6735K1 \) is divisible by 9, we will use the divisibility rule for 9. According to this rule, a number is divisible by 9 if the sum of its digits is divisible by 9.

### Step-by-Step Solution:

1. **Identify the digits of the number**: The number is \( 6735K1 \). The digits are \( 6, 7, 3, 5, K, 1 \).

2. **Calculate the sum of the known digits**:
\[ 6 + 7 + 3 + 5 + 1 = 22 \]

3. **Include \( K \) in the sum**:
\[ \text{Sum} = 22 + K \]

4. **Set up the condition for divisibility by 9**: We want \( 22 + K \) to be divisible by 9. We can express this condition as:
\[ 22 + K \equiv 0 \mod 9 \]

5. **Calculate \( 22 \mod 9 \)**:
\[ 22 \div 9 = 2 \quad \text{(remainder 4)} \]
So, \( 22 \equiv 4 \mod 9 \).

6. **Set up the equation**: We need:
\[ 4 + K \equiv 0 \mod 9 \]
This simplifies to:
\[ K \equiv -4 \mod 9 \]
or equivalently:
\[ K \equiv 5 \mod 9 \]

7. **Determine the least value of \( K \)**: The smallest non-negative integer satisfying \( K \equiv 5 \mod 9 \) is \( K = 5 \).

### Conclusion:
The least value of \( K \) such that \( 6735K1 \) is divisible by 9 is \( K = 5 \).

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