`vec(a)` and `vec(b)` are unit vectors and angle between them is `pi/k`. If `vec(a)+2vec(b)` and `5vec(a)-4vec(b)` are perpendicular to each other then find the integer value of k.

Explanation

To solve the problem, we need to find the integer value of \( k \) given that \( \vec{a} \) and \( \vec{b} \) are unit vectors, the angle between them is \( \frac{\pi}{k} \), and the vectors \( \vec{a} + 2\vec{b} \) and \( 5\vec{a} - 4\vec{b} \) are perpendicular.

### Step-by-Step Solution:

1. **Understand the Condition of Perpendicularity**:
Two vectors are perpendicular if their dot product is zero. Therefore, we can write:
\[ (\vec{a} + 2\vec{b}) \cdot (5\vec{a} - 4\vec{b}) = 0 \]

2. **Expand the Dot Product**:
Expanding the left-hand side:
\[ \vec{a} \cdot (5\vec{a}) + \vec{a} \cdot (-4\vec{b}) + 2\vec{b} \cdot (5\vec{a}) + 2\vec{b} \cdot (-4\vec{b}) \]
This simplifies to:
\[ 5(\vec{a} \cdot \vec{a}) - 4(\vec{a} \cdot \vec{b}) + 10(\vec{b} \cdot \vec{a}) - 8(\vec{b} \cdot \vec{b}) \]

3. **Substitute Values for Unit Vectors**:
Since \( \vec{a} \) and \( \vec{b} \) are unit vectors:
\[ \vec{a} \cdot \vec{a} = 1 \quad \text{and} \quad \vec{b} \cdot \vec{b} = 1 \]
Thus, we can rewrite the equation:
\[ 5(1) - 4(\vec{a} \cdot \vec{b}) + 10(\vec{a} \cdot \vec{b}) - 8(1) = 0 \]
Simplifying gives:
\[ 5 - 8 + 6(\vec{a} \cdot \vec{b}) = 0 \]
Which leads to:
\[ -3 + 6(\vec{a} \cdot \vec{b}) = 0 \]

4. **Solve for \( \vec{a} \cdot \vec{b} \)**:
Rearranging gives:
\[ 6(\vec{a} \cdot \vec{b}) = 3 \quad \Rightarrow \quad \vec{a} \cdot \vec{b} = \frac{1}{2} \]

5. **Relate the Dot Product to the Angle**:
The dot product of two vectors can also be expressed in terms of the cosine of the angle \( \theta \) between them:
\[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta \]
Since both vectors are unit vectors, this simplifies to:
\[ \vec{a} \cdot \vec{b} = \cos\theta \]
Therefore:
\[ \cos\theta = \frac{1}{2} \]

6. **Find the Angle**:
The angle \( \theta \) corresponding to \( \cos\theta = \frac{1}{2} \) is:
\[ \theta = \frac{\pi}{3} \]

7. **Relate to Given Angle**:
We know from the problem statement that the angle between \( \vec{a} \) and \( \vec{b} \) is also given by:
\[ \theta = \frac{\pi}{k} \]
Setting these equal gives:
\[ \frac{\pi}{k} = \frac{\pi}{3} \]

8. **Solve for \( k \)**:
By cross-multiplying, we find:
\[ k = 3 \]

### Final Answer:
The integer value of \( k \) is \( \boxed{3} \).