Learn Hardcore Japanese: denwabangou wo matigaezuni okutta node, henzi ga sugu kita. - Because I sent the phone number without mistakes, the reply came quickly.

電話番号を marks the object “phone number.”
間違えずに is the adverbial form of the negative verb 間違える (“to make a mistake”), built by taking the 未然形 stem 間違え and adding ずに, meaning “without making a mistake.”
送った is the past tense of 送る (“to send”).
Altogether it means “(I) sent the phone number without making any mistakes.”

Why is ずに used instead of ないで in 間違えずに?

Both 〜ずに and 〜ないで can mean “without doing ~.”
– 〜ずに is somewhat more formal or written.
– 〜ないで is more colloquial.
In this sentence, 間違えずに sounds concise and slightly more formal, fitting a written or polite context.

What role does ので play in 間違えずに送ったので、返事がすぐ来た?

ので is a conjunction meaning “because” or “since,” used to link a cause or reason to a result.
Here, 間違えずに送ったので means “because (I) sent it without mistakes,” and it introduces the outcome 返事がすぐ来た (“the reply came quickly”).

How is 返事が来た different from 返事をくれた?

– 返事が来た literally means “a reply came,” focusing on the event of receiving. It’s neutral and objective.
– 返事をくれた means “(someone) gave me a reply,” highlighting the giver’s action and often implying gratitude.
In many contexts, 返事が来た is more natural when you simply want to report that a reply arrived.

What does すぐ modify in this sentence, and could you use すぐに instead?

すぐ is an adverb modifying 来た, so it describes how quickly the reply arrived (“immediately”).
Yes, you could also say すぐに来た, which is equally correct. The に is optional with some adverbs like すぐ; leaving it off often sounds a bit more casual.

Why is the particle を used after 電話番号 but が used after 返事?

– を marks the direct object of the verb 送った (“sent”), so 電話番号を送った = “sent the phone number.”
– が marks the subject of the intransitive verb 来た (“came”), so 返事が来た = “the reply came.”

Could you rewrite this sentence more politely or formally?

Yes. For example:
「電話番号を間違えずに送信したので、すぐに返信が届きました。」
– 送信する is a more formal verb for “to send.”
– 返信 is a slightly more formal term for “reply.”
– 届きました is the polite past form of 届く (“to arrive”).