Alright! Let's break down the problem and find the smallest perfect square that is exactly divisible by 2 and 3.
1. Understand the criteria:
- A number that is a perfect square.
- That perfect square must be divisible by both 2 and 3.
2. Analyze the factors of a perfect square:
- A perfect square is a number that can be expressed as [tex]\( k^2 \)[/tex] for some integer [tex]\( k \)[/tex].
- If [tex]\( k^2 \)[/tex] is divisible by 2 and 3, then [tex]\( k \)[/tex] must be divisible by the least common multiple (LCM) of 2 and 3.
3. Find the LCM of 2 and 3:
- The LCM of 2 and 3 is 6 because it's the smallest number that 2 and 3 both divide into without leaving a remainder.
4. Determine the smallest k that satisfies k is divisible by 6:
- Since [tex]\( k \)[/tex] must be divisible by 6, the smallest such [tex]\( k \)[/tex] is 6 itself.
5. Calculate the perfect square:
- [tex]\( k^2 = 6^2 = 36 \)[/tex].
So, the smallest perfect square that is divisible by both 2 and 3 is [tex]\( 36 \)[/tex].
Next, let's discuss the quadrilaterals whose diagonals are perpendicular:
- Rhombus: In a rhombus, all sides are equal in length, and the diagonals bisect each other at right angles (90 degrees).
- Square: A square is a special type of rhombus where all the interior angles are right angles (90 degrees). Thus, its diagonals also bisect each other at right angles.
- Kite: In a kite, two pairs of adjacent sides are of equal length, and the diagonals intersect at right angles. One diagonal bisects the other.
To summarize:
- To find the smallest perfect square divisible by 2 and 3, the smallest such number is 36.
- Quadrilaterals whose diagonals are perpendicular include the rhombus, square, and kite.
