If 60 is a factor of u, is 18 a factor of u?

Answer

To determine if 18 is a factor of \( u \) given that 60 is a factor of \( u \), we can follow these steps:

### Step 1: Understand the relationship between factors and divisibility
Since 60 is a factor of \( u \), we can express \( u \) as:
\[ u = 60n \]
where \( n \) is an integer.

### Step 2: Check if 18 is a factor of \( u \)
To check if 18 is a factor of \( u \), we need to see if \( u \) is divisible by 18. This means we need to check if:
\[ \frac{u}{18} = \frac{60n}{18} \]
is an integer.

### Step 3: Simplify the expression
We simplify \( \frac{60n}{18} \):
\[ \frac{60n}{18} = \frac{60}{18} \cdot n = \frac{10}{3} \cdot n \]
This means that \( \frac{60n}{18} \) will be an integer if \( n \) is a multiple of 3.

### Step 4: Analyze different values of \( n \)
1. If \( n = 1 \):
\[ \frac{60 \cdot 1}{18} = \frac{60}{18} = \frac{10}{3} \quad \text{(not an integer)} \]
Hence, \( u \) is not divisible by 18.

2. If \( n = 3 \):
\[ \frac{60 \cdot 3}{18} = \frac{180}{18} = 10 \quad \text{(is an integer)} \]
Hence, \( u \) is divisible by 18.

3. If \( n = 6 \):
\[ \frac{60 \cdot 6}{18} = \frac{360}{18} = 20 \quad \text{(is an integer)} \]
Hence, \( u \) is divisible by 18.

### Step 5: Conclusion
Since \( u \) can be expressed as \( 60n \) for any integer \( n \), and we found cases where \( u \) is both divisible and not divisible by 18, we conclude that we cannot definitively say whether 18 is a factor of \( u \) without knowing the specific value of \( n \).

### Final Answer
We cannot determine if 18 is a factor of \( u \) based on the information given.

---