Questions: Which of the following partial order diagrams is the correct diagram for the divisor order on D24 := 1, 2, 3, 4, 6, 8, 12, 24 ordered by divisibility?

△ Identify the correct Hasse diagram for the set \( D_{24} = \{1, 2, 3, 4, 6, 8, 12, 24\} \) ordered by divisibility. ○ Determine prime factorizations ☼ The prime factorizations are: \( 1 = 1 \), \( 2 = 2 \), \( 3 = 3 \), \( 4 = 2^2 \), \( 6 = 2 \times 3 \), \( 8 = 2^3 \), \( 12 = 2^2 \times 3 \), \( 24 = 2^3 \times 3 \). ○ Establish divisibility relations ☼ The divisibility relations are: \( 1 \mid 2, 3 \); \( 2 \mid 4, 6 \); \( 3 \mid 6 \); \( 4 \mid 8, 12 \); \( 6 \mid 12 \); \( 8 \mid 24 \); \( 12 \mid 24 \). ○ Construct Hasse diagram layers ☼ Layers are:

• Layer 0: 1
• Layer 1: 2, 3
• Layer 2: 4, 6
• Layer 3: 8, 12
• Layer 4: 24 ○ Verify connections in the provided diagram ☼ The provided diagram correctly shows:
• 24 connected to 12, 8
• 12 connected to 4, 6
• 8 connected to 4
• 6 connected to 2, 3
• 4 connected to 2
• 2 and 3 connected to 1 (implied) ✧ The provided diagram is the correct Hasse diagram for the set \( D_{24} = \{1, 2, 3, 4, 6, 8, 12, 24\} \) ordered by divisibility. ☺The provided diagram is the correct Hasse diagram for the set $D_{24} = \{1, 2, 3, 4, 6, 8, 12, 24\}$ ordered by divisibility.