Understanding Divisibility by 22
To determine if the seven-digit number \( N = 47a342b \) is divisible by 22, it must satisfy the conditions for divisibility by both 2 and 11.

Condition for Divisibility by 2
- A number is divisible by 2 if its last digit is even.
- This means \( b \) must be one of the following values: 0, 2, 4, 6, or 8.

Condition for Divisibility by 11
- A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or divisible by 11.
- For \( N = 47a342b \):
- Odd position digits: \( 4, 7, 3, b \) (sum = \( 4 + 7 + 3 + b = 14 + b \))
- Even position digits: \( 7, a, 2 \) (sum = \( 7 + a + 2 = 9 + a \))
- The difference is:
\[
|(14 + b) - (9 + a)| = |5 + b - a|
\]
- For \( N \) to be divisible by 11, \( |5 + b - a| \) must equal 0 or be divisible by 11.

Evaluating Possible Values
- **Case 1:** \( 5 + b - a = 0 \)
\[
a = b + 5
\]
Possible values for \( b \) are 0, 2, 4, 6, 8, resulting in \( a \) values 5, 7, 9, 11, 13, respectively. Only valid \( a \) is 5, 7, 9.
- **Case 2:** \( 5 + b - a = 11 \)
\[
a = b - 6
\]
Possible values for \( b \) lead to \( a \) values of -6, -4, -2, 0, 2, which are invalid.
- **Case 3:** \( 5 + b - a = -11 \)
\[
a = b + 16
\]
This case yields no valid \( a \) values since \( b \) is limited to even digits.

Conclusion
The valid pairs \((a, b)\) are:
- \( (5, 0) \)
- \( (7, 2) \)
- \( (9, 4) \)
Thus, there are **3 possible values** for \( N \) that satisfy the divisibility condition by 22.