The $n$th statement in a list of $100$ statements is "Exactly $n$ of the statements in this list are false".

Your list is:

• Exactly 1 statement in this list is false.
• Exactly 2 statements in this list are false.
• $\vdots$
• Exactly $n$ statements in this list are false.
• $\vdots$
• Exactly 100 statements in this list are false.

Consider: Can all the statements be false? Can more than one statement be true?

I have seen a good YouTube video explaining a problem of this sort (it can be found here by MindYourDecisions)

Essentially, consider the $100^{th}$ statement for the exact case. If it's true, then all the statements, including itself, must be false. This leads to a contradiction and it must be false. We now know that least one statement is false

All these statements are mutually exclusive, and therefore only one of them can be true at a time. So that would mean $99$ of them would be false. Hence, $99^{th}$ statement must be true.

I'll let you now use this to try and figure out the "at least" case.

Credits to MindYourDecisions for the video solution