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The problems provided in the image are as follows:

1. **Writing 120 as a product of its prime factors.**
2. **Finding the smallest integer value of \( n \) such that \( 315n \) is a square number.**

Let's solve these problems one by one.

### Part (a): Prime Factorization of 120
To write 120 as a product of its prime factors, we perform repeated division by prime numbers:
1. \( 120 \div 2 = 60 \)
2. \( 60 \div 2 = 30 \)
3. \( 30 \div 2 = 15 \)
4. \( 15 \div 3 = 5 \)
5. \( 5 \div 5 = 1 \)

Thus, \( 120 = 2^3 \times 3 \times 5 \).

### Part (b): Finding \( n \) such that \( 315n \) is a square number
The prime factorization of 315 is given as \( 315 = 3^2 \times 5 \times 7 \).
For \( 315n \) to be a square number, the exponent of each prime factor in the product must be even. Currently, \( 3 \) has an even exponent (2), but both \( 5 \) and \( 7 \) have odd exponents (1). 

To make the exponents even, we need:
- One more factor of \( 5 \) to make the exponent of \( 5 \) equal to 2.
- One more factor of \( 7 \) to make the exponent of \( 7 \) equal to 2.

Therefore, \( n \) must at least include \( 5 \) and \( 7 \) to balance the exponents:
\[ n = 5 \times 7 = 35 \]

So, the smallest integer \( n \) such that \( 315n \) is a square number is \( 35 \).

Would you like any further details on these solutions or have any other questions? Here are some follow-up queries you might consider:

1. What are the steps involved in prime factorization?
2. How do we determine if a number is a square number?
3. Can you explain how the exponents of prime factors influence whether a number is square?
4. What are other examples where you need to find an \( n \) that makes a product a square number?
5. How can the concept of least common multiple be applied to similar problems?

**Tip:** When factorizing a number, always start with the smallest prime number and work your way up. This method ensures all factors are prime.

The original question submitted by the user as text.

Explore how to determine the smallest integer value n such that 315n results in a square number, using prime factorization and understanding of square numbers. Suitable for students in Grades 7-9, enhancing grasp of fundamental arithmetic.

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