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To solve the problem of finding the number of positive integers (x \leq 100) not divisible by 2, 3, or 5, we use the inclusion-exclusion principle.

Step 1: Define Sets

Let:

• (A): Numbers ≤100 divisible by 2
• (B): Numbers ≤100 divisible by 3
• (C): Numbers ≤100 divisible by 5

We need to compute (100 - |A \cup B \cup C|) (total numbers minus those divisible by at least one of 2,3,5).

Step 2: Apply Inclusion-Exclusion

[|A \cup B \cup C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|]

Calculate each term:

• (|A| = \lfloor 100/2 \rfloor = 50)
• (|B| = \lfloor 100/3 \rfloor = 33)
• (|C| = \lfloor 100/5 \rfloor = 20)
• (|A∩B| = \lfloor 100/6 \rfloor = 16) (divisible by LCM(2,3)=6)
• (|A∩C| = \lfloor 100/10 \rfloor = 10) (divisible by LCM(2,5)=10)
• (|B∩C| = \lfloor 100/15 \rfloor = 6) (divisible by LCM(3,5)=15)
• (|A∩B∩C| = \lfloor 100/30 \rfloor = 3) (divisible by LCM(2,3,5)=30)

Step 3: Compute Result

[|A \cup B \cup C| = 50 + 33 + 20 - 16 -10 -6 +3 = 74]

[100 - 74 = 26]

Answer: (\boxed{26})

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