We are given that the lowest common multiple (LCM) of $y$ and 120 is 360. We are tasked with finding the smallest possible value of $y$.

Step 1: Prime Factorization of 120 and 360

First, we find the prime factorizations of 120 and 360.

• $120 = 2^3 \times 3 \times 5$
• $360 = 2^3 \times 3^2 \times 5$

Step 2: Use of LCM Formula

The formula for the LCM of two numbers is the product of the highest powers of all the prime factors that appear in the factorizations of both numbers.

We know: $\text{LCM}(y, 120) = 360$

This means that the prime factorization of $y$ must combine with the prime factorization of 120 to produce the prime factorization of 360.

Step 3: Analyze Prime Factors for $y$

Since $360 = 2^3 \times 3^2 \times 5$ and $120 = 2^3 \times 3 \times 5$, we compare them to find out what factors $y$ must include:

• For the prime factor $2$, $120$ already has $2^3$, and $360$ also has $2^3$. Therefore, $y$ does not need any additional factors of 2.
• For the prime factor $3$, $120$ has $3^1$, but $360$ has $3^2$. This means that $y$ must include at least one more factor of 3 to contribute to the total of $3^2$.
• For the prime factor $5$, $120$ already has $5$, and $360$ also has $5$, so $y$ does not need any additional factors of 5.

Thus, the smallest $y$ must have $3^1$ as its prime factor to account for the missing factor of 3 in 120. Therefore, the smallest value of $y$ is: $y = 3$

Final Answer:

The smallest possible value of $y$ is $\boxed{3}$.

Would you like a further breakdown or have any questions?

Here are 5 related questions for further exploration:

1. What is the LCM of 18 and 24?
2. How do you find the greatest common divisor (GCD) of two numbers?
3. Can two numbers have the same LCM and GCD?
4. What is the relation between LCM and GCD of two numbers?
5. How do you find the LCM of three or more numbers?

Tip: The LCM can be helpful in problems involving synchronization of cycles or periodic events.