A math problem:

There are students who are to share computers. Each student uses exactly computer and no computer is used by more than students. Show that at least computers are used by or more students.

One kind of proof using an argument by contradiction:

Suppose not. Suppose that or fewer computers are used by or more students.[A contradiction will be derived.]Then or more computers are used by or fewer students.

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(End of first proof)

My question 1:

How did the author deduce that "Then or more computers are used by or fewer students"?

Another kind of proof using a direct argument:

Let be the number of computers used by or more students.[We must show that ] Because each computer is used by at most students, these computers are used by at most students(by the contrapositive form of the generalized pigeonhole principle). The remaining computers are each used by at most students.

.........

(End of this proof)

Another question 2:

Why's that "The remaining computers are each used by at most students."?