Vertices Logic: Identifying The Figure That Doesn't Belong

This problem requires us to identify the figure that is different from the others based on its number of vertices. We need to carefully examine each shape presented.

Analyzing Each Figure Based on Vertices

We will look at each option, identify the shape, and count its vertices. A vertex is a corner point where edges meet.

Option	Figure Type	Image Representation	Number of Vertices	Divisibility Check (by 2 or 3)
1	Pentagon		5	No ($5 \pmod{2} \neq 0$ and $5 \pmod{3} \neq 0$)
2	Hexagon		6	Yes ($6 \pmod{2} = 0$ and $6 \pmod{3} = 0$)
3	Square		4	Yes ($4 \pmod{2} = 0$)
4	Triangle		3	Yes ($3 \pmod{3} = 0$)

Identifying the Pattern Based on Vertex Count

Let's list the number of vertices for each shape:

• Option 1: 5 vertices
• Option 2: 6 vertices
• Option 3: 4 vertices
• Option 4: 3 vertices

We need to find a pattern or property that connects three of the figures, making the fourth one the outlier. Let's consider the divisibility of the number of vertices ($n$) by the numbers 2 and 3.

• For $n=3$ (Triangle): It is divisible by 3. ($3 \pmod{3} = 0$)
• For $n=4$ (Square): It is divisible by 2. ($4 \pmod{2} = 0$)
• For $n=5$ (Pentagon): It is not divisible by 2 and not divisible by 3. ($5 \pmod{2} \neq 0$, $5 \pmod{3} \neq 0$)
• For $n=6$ (Hexagon): It is divisible by 2 and divisible by 3. ($6 \pmod{2} = 0$, $6 \pmod{3} = 0$)

We can see that the number of vertices for the Triangle (3), Square (4), and Hexagon (6) are all divisible by 2 or 3.

Conclusion: The Figure That Doesn't Belong

The Pentagon (Option 1) has 5 vertices. Since 5 is not divisible by 2 or 3, it stands out from the other figures whose vertex counts (3, 4, and 6) are divisible by 2 or 3. Therefore, the Pentagon is the figure that doesn't belong based on this vertex divisibility pattern.