For what value of N, 23N7 is completely divisible by 9?

Options

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

Answer

To determine the value of \( N \) in the number \( 23N7 \) such that it is completely divisible by 9, we can follow these steps:

### Step 1: Understand the divisibility rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

### Step 2: Identify the digits of the number
The digits of the number \( 23N7 \) are \( 2, 3, N, \) and \( 7 \).

### Step 3: Calculate the sum of the known digits
First, we calculate the sum of the known digits:
\[ 2 + 3 + 7 = 12 \]
Now, we include \( N \) in the sum:
\[ \text{Sum} = 12 + N \]

### Step 4: Set up the condition for divisibility by 9
We need \( 12 + N \) to be divisible by 9. This can be expressed as:
\[ 12 + N \equiv 0 \ (\text{mod} \ 9) \]

### Step 5: Calculate \( 12 \mod 9 \)
Calculating \( 12 \mod 9 \):
\[ 12 \equiv 3 \ (\text{mod} \ 9) \]
Thus, we need:
\[ 3 + N \equiv 0 \ (\text{mod} \ 9) \]
This simplifies to:
\[ N \equiv -3 \equiv 6 \ (\text{mod} \ 9) \]

### Step 6: Determine the possible values for \( N \)
The value of \( N \) must be a single digit (0-9). The only value that satisfies \( N \equiv 6 \ (\text{mod} \ 9) \) is:
\[ N = 6 \]

### Conclusion
Thus, the value of \( N \) that makes \( 23N7 \) divisible by 9 is:
\[ \boxed{6} \]