I'm going to be here for exactly 10 minutes. The position of adverb "exactly".

Hi all,
I have just seen this sentence on a website.
I'm going to be here for exactly 10 minutes.

I am confused about the use of the adverb "exactly" here. I always think that adverbs's position in a sentence as follow:
1) initial position (before the subject)
2) mid position (after or before verb )
3) end position
However, in the above sentence the adverb "exactly" is preceded by a preposition. I have never seen this before.
Could you tell me what verb or adverb or adjective that "exactly" modifies?
I think that in this case "10" is an adjective and this sentence is OK. Is it right?
But what part that "exactly" modifies in these sentences:
1. In fact, I'm in the process of getting clearance to sell them for exactly that purpose.
2. What do you need money for exactly?
3. He saw this emotion for exactly what it was.
Thank you.

Numbers are usually adjectives because they give information about nouns. Therefore, you think correctly.
Remember adverbs can add to the meaning of the whole sentence. You do not necessarily need to search for just a word to refer it to.

Thank you.
Could you tell me about other sentences when there are no adjective that "exactly" modifies?
Then how can I explain for their positions?
For example, for exactly that purpose, for exactly, for exactly what it was.
(I can only think it is appropriate if I consider "that" as an adjective.)

It's pretty clear to me that it is "ten" that "exactly" modifies. He's not going to be there for 9 minutes or 11 minutes. But for exactly ten minutes.