We can rewrite the given equation as $(2 \times 11)^2 \times (5 \times 11)^2 = (2 \times 5)^2 \times N^2$.
Simplifying both sides, we have $2^2 \times 11^2 \times 5^2 \times 11^2 = 2^2 \times 5^2 \times N^2$.
Cancelling out the common factors, we get $11^2 \times 11^2 = N^2$.
Taking the square root of both sides, we have $11 \times 11 = N$.
Therefore, $N = \boxed{121}$.
The answer is: 121
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$

### Answer:
