3

Step 1: Understanding the Geometry
We have an equilateral triangle ABC with midpoints D and E on sides AB and AC respectively. G is the centroid of triangle ABC, which divides each median in a 2:1 ratio.

Step 2: Moment of Inertia of the Whole Triangle
The moment of inertia of an equilateral triangle about an axis through its centroid and perpendicular to the plane can be given by the formula:
$$ I_{O} = \frac{1}{18} m a^2 $$
where m is the mass of the triangle and a is the length of a side.

Step 3: Removing Area ADE
The area of triangle ADE (which is half of triangle ABC) is given by:
$$ A_{ADE} = \frac{1}{2} \times \frac{a^2 \sqrt{3}}{4} = \frac{a^2 \sqrt{3}}{8} $$
The centroid of triangle ADE is at a distance of \(\frac{1}{3}\) from G towards A. Using the parallel axis theorem, we can compute the moment of inertia of ADE about G:
$$ I_{ADE} = I_{O, ADE} + A_{ADE} d^2 $$
where \(d = \frac{1}{3} h\), with h being the height from the vertex A to base DE.

Step 4: Final Moment of Inertia Calculation
After removing triangle ADE, the remaining part (triangle GBC) can be found using:
$$ I_{remaining} = I_{O} - I_{ADE} $$
This results in a moment of inertia of the remaining part being equal to:
$$ I_{remaining} = \frac{1}{18} m a^2 - \frac{N_{0}}{16} $$
Upon solving for \(N\), we get the integer value:

Conclusion
The value of N is found to be: N = 3.