To solve the problem, we need to use principles of set theory and inclusion-exclusion. Let's denote:

• $n = 10,000$, the total range of integers.
• $A$: Integers divisible by 3.
• $B$: Integers divisible by 5.
• $C$: Integers divisible by 7.
• $D$: Integers divisible by 11.

We are tasked to find integers divisible by both 3 and 5 (i.e., $15$) but not divisible by $7$ or $11$. Mathematically:

$|A \cap B \setminus (C \cup D)|$

Step 1: Total numbers divisible by 15

The numbers divisible by 15 are:

$|A \cap B| = \left\lfloor \frac{n}{15} \right\rfloor = \left\lfloor \frac{10,000}{15} \right\rfloor = 666$

Step 2: Numbers divisible by both 15 and 7

The least common multiple (LCM) of $15$ and $7$ is $105$. The count is:

$|A \cap B \cap C| = \left\lfloor \frac{n}{105} \right\rfloor = \left\lfloor \frac{10,000}{105} \right\rfloor = 95$

Step 3: Numbers divisible by both 15 and 11

The LCM of $15$ and $11$ is $165$. The count is:

$|A \cap B \cap D| = \left\lfloor \frac{n}{165} \right\rfloor = \left\lfloor \frac{10,000}{165} \right\rfloor = 60$

Step 4: Numbers divisible by 15, 7, and 11

The LCM of $15$, $7$, and $11$ is $1155$. The count is:

$|A \cap B \cap C \cap D| = \left\lfloor \frac{n}{1155} \right\rfloor = \left\lfloor \frac{10,000}{1155} \right\rfloor = 8$

Step 5: Inclusion-Exclusion to find $|A \cap B \cap (C \cup D)|$

The numbers divisible by $15$ and $7$ or $11$ are:

$|A \cap B \cap (C \cup D)| = |A \cap B \cap C| + |A \cap B \cap D| - |A \cap B \cap C \cap D|$

Substitute the values:

$|A \cap B \cap (C \cup D)| = 95 + 60 - 8 = 147$

Step 6: Subtract from total numbers divisible by 15

The count of integers divisible by $15$ but not $7$ or $11$ is:

$|A \cap B \setminus (C \cup D)| = |A \cap B| - |A \cap B \cap (C \cup D)|$

Substitute the values:

$|A \cap B \setminus (C \cup D)| = 666 - 147 = 519$

Final Answer:

The number of integers between 1 and 10,000 inclusive that are divisible by 3 and 5 but not by 7 or 11 is:

$\boxed{519}$

Let me know if you want a detailed breakdown or additional examples! 😊

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1. How does inclusion-exclusion work in set problems?
2. Can we generalize this approach to other divisors?
3. What is the importance of calculating LCM in such problems?
4. How do you interpret modular arithmetic in divisibility rules?
5. What if the range was extended to a different number, say 50,000?

Tip: For large ranges, writing modular rules as functions helps automate the calculations!