To solve this, we use the principle of inclusion-exclusion. We need to count the integers between 1 and 10,000 that are divisible by at least two of the numbers $3$, $5$, $7$, and $11$.

Step-by-step Solution:

1. Define the sets: Let $A$, $B$, $C$, and $D$ represent the sets of integers divisible by $3$, $5$, $7$, and $11$, respectively.

2. Inclusion-Exclusion Formula: To find the count of integers divisible by at least two of these numbers, we subtract the counts of integers divisible by one number alone and add back the overlaps of three or more: $|A \cap B \cap C \cap D| = \sum_{i<j} |A_i \cap A_j| - \sum_{i<j<k} |A_i \cap A_j \cap A_k| + |A \cap B \cap C \cap D|$

3. Key Calculations: For each divisor $d$, count how many integers between $1$ and $10,000$ are divisible by $d$: $\text{Count divisible by } d = \left\lfloor \frac{10,000}{d} \right\rfloor$

   • Integers divisible by 3: $\lfloor 10,000 / 3 \rfloor = 3333$
   • Integers divisible by 5: $\lfloor 10,000 / 5 \rfloor = 2000$
   • Integers divisible by 7: $\lfloor 10,000 / 7 \rfloor = 1428$
   • Integers divisible by 11: $\lfloor 10,000 / 11 \rfloor = 909$

   Now, for pairs, triples, and quadruples of divisors, compute the least common multiple (LCM) and divide $10,000$ by that LCM:

   For pairs:

   • Divisible by $\text{lcm}(3, 5) = 15$: $\lfloor 10,000 / 15 \rfloor = 666$
   • Divisible by $\text{lcm}(3, 7) = 21$: $\lfloor 10,000 / 21 \rfloor = 476$
   • Divisible by $\text{lcm}(3, 11) = 33$: $\lfloor 10,000 / 33 \rfloor = 303$
   • Divisible by $\text{lcm}(5, 7) = 35$: $\lfloor 10,000 / 35 \rfloor = 285$
   • Divisible by $\text{lcm}(5, 11) = 55$: $\lfloor 10,000 / 55 \rfloor = 181$
   • Divisible by $\text{lcm}(7, 11) = 77$: $\lfloor 10,000 / 77 \rfloor = 129$

   For triples:

   • Divisible by $\text{lcm}(3, 5, 7) = 105$: $\lfloor 10,000 / 105 \rfloor = 95$
   • Divisible by $\text{lcm}(3, 5, 11) = 165$: $\lfloor 10,000 / 165 \rfloor = 60$
   • Divisible by $\text{lcm}(3, 7, 11) = 231$: $\lfloor 10,000 / 231 \rfloor = 43$
   • Divisible by $\text{lcm}(5, 7, 11) = 385$: $\lfloor 10,000 / 385 \rfloor = 25$

   For quadruples:

   • Divisible by $\text{lcm}(3, 5, 7, 11) = 1155$: $\lfloor 10,000 / 1155 \rfloor = 8$

4. Apply Inclusion-Exclusion: Using inclusion-exclusion for at least two divisors: $\text{Count} = \left( \text{sum of pairwise overlaps} \right) - \left( \text{sum of triple overlaps} \right) + \left( \text{count of quadruple overlap} \right)$ Substitute the values: $\text{Count} = (666 + 476 + 303 + 285 + 181 + 129) - (95 + 60 + 43 + 25) + 8$ Simplify: $\text{Count} = 2040 - 223 + 8 = 1825$

Final Answer:

The number of integers between $1$ and $10,000$ that are divisible by at least two of $3$, $5$, $7$, and $11$ is 1825.

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Would you like further details or any clarifications? Here are 5 related questions to expand your understanding:

1. How is the principle of inclusion-exclusion derived, and why does it work?
2. What are some examples where inclusion-exclusion is applied beyond number theory?
3. How do we compute the least common multiple (LCM) efficiently for multiple numbers?
4. What would change if we were considering "at least three" divisors instead of two?
5. Can this approach be generalized for more divisors or a larger range?

Tip: Familiarize yourself with the inclusion-exclusion principle, as it is widely applicable in probability, combinatorics, and number theory!