Consider the number `N = 8 7 a 2 7 9 3 1 b` , where a , b are single digit whole numbers.
Number of possible ordred pairs (a, b) for which N is divisible by 72 is `"(a) 0 (b) 1 (c) 2 (d) 3"`

Options

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

Explanation

To determine the ordered pairs (a, b) such that the number \( N = 87a27931b \) is divisible by 72, we need to ensure that \( N \) is divisible by both 8 and 9.

### Step 1: Check divisibility by 8
For a number to be divisible by 8, its last three digits must be divisible by 8. The last three digits of \( N \) are \( 31b \).

To find the values of \( b \) that make \( 31b \) divisible by 8, we can check each possible digit (0-9) for \( b \):

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

From this, we find that the only value for \( b \) that makes \( 31b \) divisible by 8 is \( b = 2 \).

### Step 2: Check divisibility by 9
Next, we need to ensure that \( N \) is divisible by 9. For this, the sum of its digits must be divisible by 9.

The sum of the digits in \( N \) is:
\[ 8 + 7 + a + 2 + 7 + 9 + 3 + 1 + b \]
Substituting \( b = 2 \):
\[ 8 + 7 + a + 2 + 7 + 9 + 3 + 1 + 2 = 39 + a \]

Now we need \( 39 + a \) to be divisible by 9. We can check the possible values of \( a \) (0-9):

- \( a = 0 \): \( 39 + 0 = 39 \) (not 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 values of \( a \) that make \( 39 + a \) divisible by 9 are \( a = 3, 6, 9 \).

### Conclusion
Thus, the ordered pairs \( (a, b) \) that satisfy both conditions are:
1. \( (3, 2) \)
2. \( (6, 2) \)
3. \( (9, 2) \)

Therefore, the number of possible ordered pairs \( (a, b) \) for which \( N \) is divisible by 72 is **3**.