He asked me a question.

He asked a question of me.

He begged me a question.
He begged a question of me.

He inquired me a question.
He inquired me of a question.

(Are the pairs all correct and interchangable? Doe they have the same meaning?)

The first pair is correct and interchangeable, but the others aren't.

The 3rd sentence is not correct.

The 5th and 6th are not correct. For this, you'd need to say, "He inquired about .... " (The "me" part wouldn't be there.)

3. He begged me a question.

4. He begged a question of me.

5. He inquired me a question.
6. He inquired a question of me.

(Thank you for your help. #6 should be changed. Among the four sentences, which ones are not grammatical? Is #6 correct?)

Among these four sentences, only #4 is correct. The word "inquire" is not used with an indirect object or "to" and an object.

He inquired about my illness.
He inquired about their children.
He inquired about the train fare from Washington to New York.

The pairs you provided are not all correct or interchangeable in terms of meaning. Let's break them down:

1. "He asked me a question."
This sentence is grammatically correct and commonly used. It means that someone posed a question to you.

2. "He asked a question of me."
This sentence is also grammatically correct and has the same meaning as the first pair. The preposition "of" indicates that the person directed the question towards you.

3. "He begged me a question."
This sentence is not correct in standard English. The verb "beg" usually takes a direct object but not followed by an indirect object in the same way as "ask." So, "He begged a question of me" would be the correct construction. However, this phrase is not commonly used and may sound unnatural.

4. "He inquired me a question."
This sentence is also not correct in standard English. It should be "He inquired of me a question." The verb "inquire" takes a preposition before the indirect object, so the correct construction would be "He inquired of me" or "He inquired of me about a question." However, this phrasing is less common than the first pair and may sound slightly formal.

To summarize, the first pair ("He asked me a question" and "He asked a question of me") are the most natural and commonly used, while the other two pairs ("He begged a question of me" and "He inquired of me a question") are less common and may sound awkward or unnatural to native English speakers.