1. **Problem Statement:** We have an undirected graph $G_c$ with vertices $\{2,3,4,5,6,7,8,9,10\}$ and edges defined between vertices whose greatest common divisor (gcd) is greater than 1. 2. **Vertices and Edges:** - Vertices: $\{2,3,4,5,6,7,8,9,10\}$ - Edges (where $\gcd(u,v) > 1$): $$(2,4), (2,6), (2,8), (2,10), (3,6), (3,9), (4,6), (4,8), (4,10), (5,10), (6,8), (6,9), (6,10), (8,10)$$ 3. **Isolated Vertex:** Vertex $7$ has no edges connecting it to any other vertex because $\gcd(7, x) = 1$ for all other vertices $x$ in the set. 4. **Connected Components:** - One connected component includes vertices $\{2,3,4,5,6,8,9,10\}$ connected by the edges above. - The vertex $7$ is isolated, forming a connected component by itself. 5. **Explanation:** - The graph is constructed by connecting vertices if their gcd is greater than 1. - This means vertices share a common factor greater than 1. - For example, $\gcd(2,4) = 2 > 1$, so edge $(2,4)$ exists. - Vertex $7$ is isolated because it is prime and shares no common factors greater than 1 with any other vertex. 6. **Summary:** The graph $G_c$ has two connected components: - Component 1: $\{2,3,4,5,6,8,9,10\}$ - Component 2: $\{7\}$ (isolated vertex)