To solve this problem, we calculate the number of integers divisible by exactly two of the divisors $3$, $5$, $7$, and $11$ in the range from $1$ to $10,000$.

This requires the principle of Inclusion-Exclusion but carefully adjusted to count only numbers divisible by exactly two of the divisors.

Step 1: Key Definitions and Inclusion-Exclusion Approach

• $A_3$: Numbers divisible by $3$.
• $A_5$: Numbers divisible by $5$.
• $A_7$: Numbers divisible by $7$.
• $A_{11}$: Numbers divisible by $11$.

The number of integers divisible by exactly two of these divisors can be obtained by:

1. Calculating the numbers divisible by any two divisors.
2. Subtracting the numbers divisible by three or four divisors.

Step 2: Direct Computations

(a) Numbers divisible by any two divisors

We compute the least common multiples (LCM) of pairs of divisors and count how many integers in $1 \leq n \leq 10,000$ are divisible by each LCM.

1. $\text{LCM}(3, 5) = 15$: $\lfloor 10,000 / 15 \rfloor = 666$.
2. $\text{LCM}(3, 7) = 21$: $\lfloor 10,000 / 21 \rfloor = 476$.
3. $\text{LCM}(3, 11) = 33$: $\lfloor 10,000 / 33 \rfloor = 303$.
4. $\text{LCM}(5, 7) = 35$: $\lfloor 10,000 / 35 \rfloor = 285$.
5. $\text{LCM}(5, 11) = 55$: $\lfloor 10,000 / 55 \rfloor = 181$.
6. $\text{LCM}(7, 11) = 77$: $\lfloor 10,000 / 77 \rfloor = 129$.

(b) Numbers divisible by three divisors

We compute the LCM of three numbers and count as above:

1. $\text{LCM}(3, 5, 7) = 105$: $\lfloor 10,000 / 105 \rfloor = 95$.
2. $\text{LCM}(3, 5, 11) = 165$: $\lfloor 10,000 / 165 \rfloor = 60$.
3. $\text{LCM}(3, 7, 11) = 231$: $\lfloor 10,000 / 231 \rfloor = 43$.
4. $\text{LCM}(5, 7, 11) = 385$: $\lfloor 10,000 / 385 \rfloor = 25$.

(c) Numbers divisible by four divisors

The LCM of $3, 5, 7, 11$ is $1155$, so:

1. $\text{LCM}(3, 5, 7, 11) = 1155$: $\lfloor 10,000 / 1155 \rfloor = 8$.

Step 3: Applying Inclusion-Exclusion for Exactly Two Divisors

The formula for numbers divisible by exactly two divisors is: $\text{Exactly Two Divisors} = \sum_{\text{pairs}} |A_{ij}| - 2 \sum_{\text{triples}} |A_{ijk}| + 3 \cdot |A_{ijkl}|$

Substituting the values: $\text{Exactly Two Divisors} = (666 + 476 + 303 + 285 + 181 + 129) - 2(95 + 60 + 43 + 25) + 3(8)$ $\text{Exactly Two Divisors} = 2040 - 446 + 24 = 1618$

Final Answer:

The number of integers between $1$ and $10,000$ divisible by exactly two of $3$, $5$, $7$, and $11$ is:

$\boxed{1618}$

Let me know if you need any clarifications or details on this solution!

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Related Questions:

1. How would you calculate the numbers divisible by exactly three of the divisors?
2. Can this method be extended to more than four divisors?
3. How is the Inclusion-Exclusion principle applied to avoid double-counting?
4. How do LCMs help in counting multiples efficiently?
5. What changes in the solution if the range is extended beyond $10,000$?

Tip:

When working with Inclusion-Exclusion, always organize your terms carefully to ensure proper alternation of addition and subtraction!