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.