Find the value of k, if `15!`=1k07674368000

Options

A. 0
B. 3
C. 2
D. 1

Answer

To find the value of \( k \) in the expression \( 15! = 1k07674368000 \), we can 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 in the number**: The number \( 1k07674368000 \) consists of the digits \( 1, k, 0, 7, 6, 7, 4, 3, 6, 8, 0, 0, 0 \).

2. **Sum the known digits**: First, we will sum all the known digits:
\[ 1 + 0 + 7 + 6 + 7 + 4 + 3 + 6 + 8 + 0 + 0 + 0 = 42 \]

3. **Include \( k \) in the sum**: Now we include \( k \) in the sum:
\[ \text{Sum} = 42 + k \]

4. **Set up the divisibility condition**: For \( 1k07674368000 \) to be divisible by 9, the total sum \( 42 + k \) must be divisible by 9.

5. **Find the remainder when 42 is divided by 9**:
\[ 42 \div 9 = 4 \quad \text{(remainder 6)} \]
This means \( 42 \equiv 6 \mod 9 \).

6. **Determine what \( k \) needs to be**: To make \( 42 + k \) divisible by 9, we need:
\[ 6 + k \equiv 0 \mod 9 \]
This simplifies to:
\[ k \equiv 3 \mod 9 \]

7. **Possible values of \( k \)**: The possible values for \( k \) that satisfy this condition are \( k = 3 \) (since \( k \) must be a single digit).

8. **Conclusion**: Therefore, the value of \( k \) is:
\[ \boxed{3} \]