Consider number N = 8 7 a 2 7 9 3 1 b, where b is a digit at unit's place and a is a digit at ten lakh's place. The greatest value of b for which N is divisible by 8 is (A9 prime (B) composite (C) even (D) odd

Options

A. `0`
B. `2`
C. `4`
D. `6`

Explanation

To determine the greatest value of \( b \) for which the number \( N = 87827931b \) is divisible by 8, we will use the divisibility rule for 8. According to this rule, a number is divisible by 8 if the number formed by its last three digits is divisible by 8.

### Step-by-Step Solution:

1. **Identify the last three digits of \( N \)**:
The last three digits of \( N \) are \( 31b \).

2. **Set up the expression for divisibility by 8**:
We need to check the values of \( 31b \) for \( b = 0, 1, 2, \ldots, 9 \) to find which values make \( 31b \) divisible by 8.

3. **Calculate \( 31b \) for each \( b \)**:
- For \( b = 0 \): \( 310 \div 8 = 38.75 \) (not divisible)
- For \( b = 1 \): \( 311 \div 8 = 38.875 \) (not divisible)
- For \( b = 2 \): \( 312 \div 8 = 39 \) (divisible)
- For \( b = 3 \): \( 313 \div 8 = 39.125 \) (not divisible)
- For \( b = 4 \): \( 314 \div 8 = 39.25 \) (not divisible)
- For \( b = 5 \): \( 315 \div 8 = 39.375 \) (not divisible)
- For \( b = 6 \): \( 316 \div 8 = 39.5 \) (not divisible)
- For \( b = 7 \): \( 317 \div 8 = 39.625 \) (not divisible)
- For \( b = 8 \): \( 318 \div 8 = 39.75 \) (not divisible)
- For \( b = 9 \): \( 319 \div 8 = 39.875 \) (not divisible)

4. **Identify the greatest value of \( b \)**:
The only value of \( b \) that makes \( 31b \) divisible by 8 is \( b = 2 \).

5. **Determine the properties of \( b \)**:
- \( b = 2 \) is a prime number.
- \( b = 2 \) is also an even number.

### Final Answer:
The greatest value of \( b \) for which \( N \) is divisible by 8 is \( b = 2 \), which is both a prime and an even number.