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Who do I need to cantact to ask about my payment problem ?

or

Whom do I need to contact to ask about my payment problem ?

Which one is correct ?
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In precise grammatical formality, "whom" is correct because it is object case.
However, this form is considered by today's standards, rather old-fashioned and quaint.
So the version with "who" is not only acceptable, it is more commonly used.
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you meant both sentences could be grammatically correct on exam paper, but the "Who" sounded more common in general conversation.

Am I correct ?
When I hear 'whom' used, it's usually incorrect, and I'd prefer that the speaker not use it at all.
However, when the pronoun is immediately preceded by a preposition--typically 'to' or 'for'--the construction is already formal enough that, to my ear, 'whom' sounds much better. Thus, "Who did you give the letter to?" is fine with me, but
*"To who did you give the letter?" sounds awful. I'd recommend "To whom did you give the letter?" and "For whom is the letter intended?".
Nor Priestyou meant both sentences could be grammatically correct on exam paper, but the "Who" sounded more common in general conversation.Am I correct ?
If your teacher is very strict, then he might mark "who" as wrong, and "whom" as correct. The formal "prescriptivist" grammar books tell us this.

But the language is changing, and some of the English grammar books do not!

The old style is called "prescriptive grammar." It sets rules, and does not look at what is really going on in the English-speaking world.
The new style is called "descriptive grammar." It changes the rules according to the way people actually speak.
There is a big argument between the prescriptivist and prescriptivist grammarians.
Students: We have free audio pronunciation exercises.