If 42573 k is divisible by 72 then the value of k is :

Options

A. 4
B. 5
C. 6
D. 7

Answer

To determine the value of \( k \) such that the number \( 42573k \) is divisible by 72, we need to check the divisibility rules for both 8 and 9, since 72 is the product of these two numbers (72 = 8 × 9).

### Step 1: Check divisibility by 8
A number is divisible by 8 if the last three digits of the number form a number that is divisible by 8. In our case, the last three digits are \( 73k \).

We need to find \( k \) such that \( 73k \) is divisible by 8.

- Possible values for \( k \) are 0 through 9. We will check each value:

1. \( k = 0 \): \( 730 \div 8 = 91.25 \) (not divisible)
2. \( k = 1 \): \( 731 \div 8 = 91.375 \) (not divisible)
3. \( k = 2 \): \( 732 \div 8 = 91.5 \) (not divisible)
4. \( k = 3 \): \( 733 \div 8 = 91.625 \) (not divisible)
5. \( k = 4 \): \( 734 \div 8 = 91.75 \) (not divisible)
6. \( k = 5 \): \( 735 \div 8 = 91.875 \) (not divisible)
7. \( k = 6 \): \( 736 \div 8 = 92 \) (divisible)
8. \( k = 7 \): \( 737 \div 8 = 92.125 \) (not divisible)
9. \( k = 8 \): \( 738 \div 8 = 92.25 \) (not divisible)
10. \( k = 9 \): \( 739 \div 8 = 92.375 \) (not divisible)

From this, we see that \( k = 6 \) makes \( 73k \) divisible by 8.

### Step 2: Check divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of \( 425736 \) (with \( k = 6 \)) is:

\[ 4 + 2 + 5 + 7 + 3 + 6 = 27 \]

Since \( 27 \div 9 = 3 \), it is divisible by 9.

### Conclusion
Since \( 425736 \) is divisible by both 8 and 9, it is divisible by 72. Therefore, the value of \( k \) is:

\[ \boxed{6} \]