Consider the number `N = 8 7 a 2 7 9 3 1 b` , where a , b are single digit whole numbers.
The least value of a for which N is is divsible by 12 is

Options

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

Explanation

To determine the least value of \( a \) for which the number \( N = 87a27931b \) is divisible by 12, we need to apply the divisibility rules for both 3 and 4, as a number is divisible by 12 if it is divisible by both of these numbers.

### Step 1: Check divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

The digits of \( N \) are \( 8, 7, a, 2, 7, 9, 3, 1, b \).

Calculating the sum of the known digits:
\[ 8 + 7 + 2 + 7 + 9 + 3 + 1 = 37 \]
Thus, the total sum becomes:
\[ 37 + a + b \]
For \( N \) to be divisible by 3, \( 37 + a + b \) must be divisible by 3.

### Step 2: Check divisibility by 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

The last two digits of \( N \) are \( 31b \). We need to check the values of \( b \) from 0 to 9 to see which values make \( 31b \) divisible by 4.

Calculating \( 31b \) for \( b = 0, 1, 2, \ldots, 9 \):
- \( b = 0 \): \( 310 \div 4 = 77.5 \) (not divisible)
- \( b = 1 \): \( 311 \div 4 = 77.75 \) (not divisible)
- \( b = 2 \): \( 312 \div 4 = 78 \) (divisible)
- \( b = 3 \): \( 313 \div 4 = 78.25 \) (not divisible)
- \( b = 4 \): \( 314 \div 4 = 78.5 \) (not divisible)
- \( b = 5 \): \( 315 \div 4 = 78.75 \) (not divisible)
- \( b = 6 \): \( 316 \div 4 = 79 \) (divisible)
- \( b = 7 \): \( 317 \div 4 = 79.25 \) (not divisible)
- \( b = 8 \): \( 318 \div 4 = 79.5 \) (not divisible)
- \( b = 9 \): \( 319 \div 4 = 79.75 \) (not divisible)

Thus, \( b \) can be either 2 or 6 for \( N \) to be divisible by 4.

### Step 3: Find the least value of \( a \)
Now we check both cases for \( b \) to find the least value of \( a \):

1. **If \( b = 2 \)**:
\[ 37 + a + 2 = 39 + a \]
We need \( 39 + a \) to be divisible by 3. The possible values for \( a \) are:
- \( a = 0 \): \( 39 + 0 = 39 \) (divisible)
- \( a = 1 \): \( 39 + 1 = 40 \) (not divisible)
- \( a = 2 \): \( 39 + 2 = 41 \) (not divisible)
- \( a = 3 \): \( 39 + 3 = 42 \) (divisible)
- \( a = 4 \): \( 39 + 4 = 43 \) (not divisible)
- \( a = 5 \): \( 39 + 5 = 44 \) (not divisible)
- \( a = 6 \): \( 39 + 6 = 45 \) (divisible)
- \( a = 7 \): \( 39 + 7 = 46 \) (not divisible)
- \( a = 8 \): \( 39 + 8 = 47 \) (not divisible)
- \( a = 9 \): \( 39 + 9 = 48 \) (divisible)

The possible values of \( a \) are \( 0, 3, 6, 9 \). The least value is \( 0 \).

2. **If \( b = 6 \)**:
\[ 37 + a + 6 = 43 + a \]
We need \( 43 + a \) to be divisible by 3. The possible values for \( a \) are:
- \( a = 0 \): \( 43 + 0 = 43 \) (not divisible)
- \( a = 1 \): \( 43 + 1 = 44 \) (not divisible)
- \( a = 2 \): \( 43 + 2 = 45 \) (divisible)
- \( a = 3 \): \( 43 + 3 = 46 \) (not divisible)
- \( a = 4 \): \( 43 + 4 = 47 \) (not divisible)
- \( a = 5 \): \( 43 + 5 = 48 \) (divisible)
- \( a = 6 \): \( 43 + 6 = 49 \) (not divisible)
- \( a = 7 \): \( 43 + 7 = 50 \) (not divisible)
- \( a = 8 \): \( 43 + 8 = 51 \) (divisible)
- \( a = 9 \): \( 43 + 9 = 52 \) (not divisible)

The possible values of \( a \) are \( 2, 5, 8 \). The least value is \( 2 \).

### Conclusion
The least value of \( a \) that makes \( N \) divisible by 12 is:
\[ \boxed{0} \]