Recall from lecture (the book does not have this incredibly standard term!) that
ID: 1890919 • Letter: R
Question
Recall from lecture (the book does not have this incredibly standard term!) that an integer n Z is "irreducible" iff the only way to write n = ab for some a, b Z is if a = 1 or b = 1. Recall also from lecture that n Z is "prime" iff whenever a, b Z and p | ab ("divides") then it must be true that p | a or p | b (or both), (in our textbook, this is not the definition, but it is at least proved as Theorem 1.5.6) In lecture we proved that if p is prime then it is also irreducible. Prove the converse: that if p is irreducible then it is also prime.Explanation / Answer
"Prove the converse"
Since you have the lecture proof of "if p is prime then it is also irreducible," simply go back to the end of the lecture proof and work backwords.
The most effective way to do this is to assume the converse:
Suppose not - that is, suppose that p is not prime. [borrow from your original prove to conclude that it must be reducible].
Use this to arrive at a contradaction - something like "p is both reducible and irreducible" or "both prime and not prime."
Therefore, you will reach a contradiction and prove the desired result.
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