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To solve the problem of finding the number of positive integers ≤1000 divisible by 3, 5, or 7, we use the inclusion-exclusion principle for three sets.

Step 1: Define the sets

Let:

• (A): Numbers ≤1000 divisible by 3
• (B): Numbers ≤1000 divisible by 5
• (C): Numbers ≤1000 divisible by 7

Step 2: Calculate individual set sizes

[ |A| = \left\lfloor \frac{1000}{3} \right\rfloor = 333 \quad (\text{since }3×333=999)
]
[ |B| = \left\lfloor \frac{1000}{5} \right\rfloor = 200 \quad (\text{since }5×200=1000)
]
[ |C| = \left\lfloor \frac{1000}{7} \right\rfloor = 142 \quad (\text{since }7×142=994)
]

Step 3: Calculate pairwise intersections

Intersections are divisible by the LCM of the two numbers:
[ |A∩B| = \left\lfloor \frac{1000}{15} \right\rfloor = 66 \quad (\text{LCM(3,5)=15; 15×66=990})
]
[ |A∩C| = \left\lfloor \frac{1000}{21} \right\rfloor = 47 \quad (\text{LCM(3,7)=21; 21×47=987})
]
[ |B∩C| = \left\lfloor \frac{1000}{35} \right\rfloor = 28 \quad (\text{LCM(5,7)=35;35×28=980})
]

Step 4: Calculate triple intersection

[ |A∩B∩C| = \left\lfloor \frac{1000}{105} \right\rfloor =9 \quad (\text{LCM(3,5,7)=105;105×9=945})
]

Step5: Apply inclusion-exclusion

[ |A∪B∪C| = |A|+|B|+|C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|
]
[ =333+200+142 -66-47-28 +9 =543
]

Answer: (\boxed{543})

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