The lowest common multiple (LCM) of x and 120 is 360.

Find the smallest possible value of x.

We are given that the lowest common multiple (LCM) of \( x \) and 120 is 360, and we need to find the smallest possible value of \( x \).

### Step 1: Prime factorization of 120 and 360
We begin by finding the prime factorizations of 120 and 360.

- \( 120 = 2^3 \times 3 \times 5 \)
- \( 360 = 2^3 \times 3^2 \times 5 \)

### Step 2: Understand the relationship between \( x \), 120, and the LCM
The LCM of two numbers is determined by taking the highest power of each prime factor that appears in either number.

Let \( x = 2^a \times 3^b \times 5^c \times p_1^{e_1} \times p_2^{e_2} \times \cdots \), where \( a, b, c \) are the powers of 2, 3, and 5 in the prime factorization of \( x \), and \( p_1, p_2, \ldots \) are other primes that do not appear in 120 or 360.

For the LCM of \( x \) and 120 to be 360, the LCM must match the prime factorization of 360. This means:

- For the factor of 2: The LCM should have \( 2^3 \). Since 120 already has \( 2^3 \), \( x \) can have \( 2^0 \) or any power up to \( 2^3 \), but it should not have more than \( 2^3 \).
- For the factor of 3: The LCM should have \( 3^2 \). Since 120 has only \( 3^1 \), \( x \) must have at least \( 3^2 \).
- For the factor of 5: The LCM should have \( 5^1 \). Since 120 already has \( 5^1 \), \( x \) can have \( 5^0 \) or \( 5^1 \), but no more than that.

### Step 3: Determine the smallest value of \( x \)
To minimize \( x \), we choose the smallest possible powers of 2 and 5, and the necessary power of 3. From the analysis:

- The smallest power of 2 is \( 2^0 \) (since 120 already has \( 2^3 \)).
- The smallest power of 3 is \( 3^2 \) (to ensure the LCM reaches \( 3^2 \)).
- The smallest power of 5 is \( 5^0 \) (since 120 already has \( 5^1 \)).

Thus, the smallest possible value of \( x \) is:
\[
x = 2^0 \times 3^2 \times 5^0 = 9
\]

### Final Answer:
The smallest possible value of \( x \) is \( \boxed{9} \).

Would you like more details, or do you have any questions?

Here are 5 related questions you might want to explore:
1. What is the prime factorization of other numbers, such as 180 or 240?
2. How do you find the greatest common divisor (GCD) of two numbers?
3. How does the LCM of three numbers differ from that of two numbers?
4. What is the relationship between the LCM and GCD of two numbers?
5. Can two numbers have an LCM smaller than both numbers?

**Tip**: When finding the LCM of two numbers, always use the highest powers of all primes present in their factorizations!

The lowest common multiple (LCM) of x and 120 is 360. Find the smallest possible value of x.

This detailed explanation guides you through finding the smallest possible value of x such that the LCM of x and 120 is 360. It uses number theory concepts like prime factorization and explores the relationship between LCM and prime powers. Suitable for Grades 8-10.

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