Given:
- The 7-digit number N = 47a342b is divisible by 22.

To Find:
- The number of possible values of N.

Explanation:

Divisibility Rule for 22:
- A number is divisible by 22 if it is divisible by both 2 and 11.
- A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
- A number is divisible by 11 if the difference between the sum of its odd-placed digits and the sum of its even-placed digits is either 0 or a multiple of 11.

Divisibility by 2:
- Since N is divisible by 22, it must be divisible by 2.
- Therefore, the last digit b must be even.

Divisibility by 11:
- To check the divisibility of N by 11, we need to find the difference between the sum of its odd-placed digits and the sum of its even-placed digits.
- The odd-placed digits in N are 4, a, 4, and b (1st, 3rd, 5th, and 7th positions).
- The even-placed digits in N are 7, 3, 2 (2nd, 4th, and 6th positions).
- The sum of the odd-placed digits = 4 + a + 4 + b.
- The sum of the even-placed digits = 7 + 3 + 2 = 12.

Case 1: Difference is 0
- If the difference between the sum of the odd-placed digits and the sum of the even-placed digits is 0, then the number is divisible by 11.
- In this case, 4 + a + 4 + b - 12 = 0.
- Simplifying, we get a + b = 4.

Case 2: Difference is a multiple of 11
- If the difference between the sum of the odd-placed digits and the sum of the even-placed digits is a multiple of 11, then the number is divisible by 11.
- In this case, 4 + a + 4 + b - 12 = 11k (where k is an integer).
- Simplifying, we get a + b = 11k + 8.

Finding Possible Values of a and b:
- From Case 1, we have a + b = 4.
- The possible values of a and b that satisfy this equation are (0, 4), (1, 3), (2, 2), (3, 1), and (4, 0) (since a and b are single digits).
- From Case 2, we have a + b = 11k + 8.
- The possible values of a and b that satisfy this equation are (3, 5), (4, 4), (5, 3), (6, 2), (7, 1), (8, 0), (9, 9), (10, 8), (11, 7), (12, 6), (13