1. He gave us a piece of paper.

2. He gave each of us a piece of paper.
3. He gave us the paper.

Does #1 mean #2?
What about #3? Is #3 correct?

The first sentence could be interpreted that he gave the group (us) one piece of paper. The second sentence makes it clear that each person was given a piece of paper. The third sentence isn't clear about how much paper he gave us. It could mean the same as either # 1 or # 2.

Thank you for your explanation.

1. Write down your name on the paper.
2. Write down your name on a paper.
3. Write down your name on a piece of paper.
4. Write down your name on the piece of paper.

I think all are correct except #2? Am I right?

# 2 could be correct if the only thing the other person wants is to see your name.

We also don't need the word "down" in these sentences. "Write your name on the paper" is the usual expression.

In the given context, #1, #2, and #3 have slightly different meanings:

1. "He gave us a piece of paper."
This sentence suggests that the person mentioned gave a single piece of paper to the group as a whole. So, everyone in the group had access to that one piece of paper.

2. "He gave each of us a piece of paper."
This sentence indicates that the person mentioned distributed multiple pieces of paper, one to each person in the group. Each individual in the group received their own separate piece of paper.

3. "He gave us the paper."
This sentence implies that there was a specific paper under discussion, and the person mentioned gave it to the group. The use of "the" implies that the paper is already known or has been referred to earlier in the conversation.

To summarize, #1 and #2 are different in terms of the quantity of paper distributed, with #1 suggesting one shared piece and #2 indicating multiple individual pieces. #3 implies a specific paper that was given to the group.