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**b. Justify \(23 \mid (460 + 92)\):**
We need to justify whether 23 divides both \(460 + 92\).

- \(460 + 92 = 552\)
- Now, check if 23 divides 552.
  \[
  552 \div 23 = 24
  \]
  Since 23 divides 552 exactly, \(23 \mid (460 + 92)\) is true.

Looking at the answer choices:
- **Option C** \(23 \mid 460\) and \(23 \mid 92\) is the correct justification. 
  - Since \(460 \div 23 = 20\) and \(92 \div 23 = 4\), 23 divides both 460 and 92.

Thus, the correct answer for part b is **C**.

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**c. Justify \(35 \mid 3^3 \cdot 5^5 \cdot 7\):**
We need to verify if 35 divides \(3^3 \cdot 5^5 \cdot 7\).

- \(35 = 5 \cdot 7\), so we need to check if both 5 and 7 are factors of \(3^3 \cdot 5^5 \cdot 7\).
  - The expression contains \(5^5\) and \(7\), so clearly both 5 and 7 divide it.

Looking at the answer choices:
- **Option B** \(35 \mid 3^3 \cdot 7\) is correct because \(35 = 5 \cdot 7\), and both 5 and 7 are factors of the given product.

Thus, the correct answer for part c is **B**.

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Would you like any further details or have any questions about these justifications?

Here are 5 related questions to expand on the topic:
1. How do you determine if one number divides another?
2. Why is it important to check individual factors when working with products like \(3^3 \cdot 5^5 \cdot 7\)?
3. Can a prime factorization method help simplify divisibility problems like this?
4. What is the general rule for determining if a number divides a sum or product?
5. How would the approach differ if the numbers were not prime factors?

**Tip:** When checking divisibility for a sum, always check if the divisor divides each term of the sum separately.

Justify 23|(460 + 92) and Justify 35|3^3 * 5^5 * 7

Explore how to justify divisibility problems, including 23 dividing the sum of 460 and 92, and 35 dividing the product of 3^3, 5^5, and 7. This problem demonstrates key number theory concepts such as divisibility and prime factorization, suitable for students in Grades 9-12.

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