Consider a number N = 2 1 P 5 3 Q 4. The number of ordered pairs (P,Q) so that the number 'N' is divisible by 44, is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To determine the number of ordered pairs \((P, Q)\) such that the number \(N = 21P53Q4\) is divisible by 44, we need to check the divisibility conditions for both 4 and 11, since \(44 = 4 \times 11\).

### Step 1: Check divisibility by 4
A number is divisible by 4 if its last two digits form a number that is divisible by 4. In our case, the last two digits are \(Q4\).

The possible values for \(Q\) can be \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\). We will check which of these values make \(Q4\) divisible by 4:

- \(04\) (divisible by 4)
- \(14\) (not divisible by 4)
- \(24\) (divisible by 4)
- \(34\) (not divisible by 4)
- \(44\) (divisible by 4)
- \(54\) (not divisible by 4)
- \(64\) (divisible by 4)
- \(74\) (not divisible by 4)
- \(84\) (divisible by 4)
- \(94\) (not divisible by 4)

Thus, the valid values for \(Q\) that make \(Q4\) divisible by 4 are:
- \(Q = 0\)
- \(Q = 2\)
- \(Q = 4\)
- \(Q = 6\)
- \(Q = 8\)

### Step 2: Check divisibility by 11
Next, we need to check the divisibility by 11. A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is a multiple of 11.

For \(N = 21P53Q4\):
- Odd positions: \(2, P, 3, 4\) → Sum = \(2 + P + 3 + 4 = P + 9\)
- Even positions: \(1, 5, Q\) → Sum = \(1 + 5 + Q = 6 + Q\)

Now, we need to find the difference:
\[ (6 + Q) - (P + 9) = Q - P - 3 \]
This difference must be a multiple of 11:
\[ Q - P - 3 = 11k \quad \text{for some integer } k \]
Rearranging gives:
\[ Q - P = 11k + 3 \]

### Step 3: Evaluate for each valid \(Q\)
Now we will evaluate the equation \(Q - P = 11k + 3\) for each valid \(Q\):

1. **For \(Q = 0\)**:
\[ 0 - P = 11k + 3 \implies -P = 11k + 3 \implies P = -11k - 3 \]
No valid \(P\) since \(P\) must be a digit (0-9).

2. **For \(Q = 2\)**:
\[ 2 - P = 11k + 3 \implies -P = 11k + 1 \implies P = -11k - 1 \]
No valid \(P\).

3. **For \(Q = 4\)**:
\[ 4 - P = 11k + 3 \implies -P = 11k - 1 \implies P = -11k + 1 \]
- For \(k = 0\): \(P = 1\) (valid)
- For \(k = 1\): \(P = -10\) (not valid)

4. **For \(Q = 6\)**:
\[ 6 - P = 11k + 3 \implies -P = 11k - 3 \implies P = -11k + 3 \]
- For \(k = 0\): \(P = 3\) (valid)
- For \(k = 1\): \(P = -8\) (not valid)

5. **For \(Q = 8\)**:
\[ 8 - P = 11k + 3 \implies -P = 11k - 5 \implies P = -11k + 5 \]
- For \(k = 0\): \(P = 5\) (valid)
- For \(k = 1\): \(P = -6\) (not valid)

### Conclusion
The valid ordered pairs \((P, Q)\) are:
1. \((1, 4)\)
2. \((3, 6)\)
3. \((5, 8)\)

Thus, the total number of ordered pairs \((P, Q)\) such that \(N\) is divisible by 44 is **3**.