Write the set by listing its elements. $$ \{x \mid x=5 n $$ for some $$ n \in N\} $$ A. $$ \{5,10,15, \ldots\} $$ B. $$ \varnothing $$ C. $$ \{10,15,20, \ldots\} $$ D. $$ \{0,5,10,15, \ldots\} $$

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To solve the problem, we need to interpret the set notation given:

$$
\{x \mid x=5n \text{ for some } n \in \mathbb{N}\}
$$

Here, $$ \mathbb{N} $$ typically represents the set of natural numbers, which usually starts from 1 (i.e., $$ \{1, 2, 3, \ldots\} $$).

1. **Understanding the expression**: The expression $$ x = 5n $$ means that $$ x $$ is a multiple of 5, where $$ n $$ is a natural number.

2. **Finding the elements**: If $$ n = 1 $$, then $$ x = 5 \times 1 = 5 $$.  
   If $$ n = 2 $$, then $$ x = 5 \times 2 = 10 $$.  
   If $$ n = 3 $$, then $$ x = 5 \times 3 = 15 $$.  
   Continuing this way, we see that the elements of the set are $$ 5, 10, 15, 20, \ldots $$.

3. **Listing the elements**: Therefore, the set can be expressed as:
   $$
   \{5, 10, 15, 20, \ldots\}
   $$

Now, let's compare this with the provided options:

A. $$ \{5,10,15, \ldots\} $$  
B. $$ \varnothing $$  
C. $$ \{10,15,20, \ldots\} $$  
D. $$ \{0,5,10,15, \ldots\} $$

The correct answer is:

**A. $$ \{5,10,15, \ldots\} $$**.
The set is $$ \{5, 10, 15, 20, \ldots\} $$, so the correct answer is A.

Write the set by listing its elements.
for some
A.
B.
C.
D.

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