We say that an integer n is prime if n ¢ {-1,0,1} and 6. for all a, b e Z, if n|ab, then n|a or n|b. Remark. We say that an integer n is irreducible if n ¢ {-1,0,1} and for all a, b E Z, if n = ab,...

can someone please show all the work for this? i have no idea how to do this

Extracted text: We say that an integer n is prime if n ¢ {-1,0,1} and 6. for all a, b e Z, if n|ab, then n|a or n|b. Remark. We say that an integer n is irreducible if n ¢ {-1,0,1} and for all a, b E Z, if n = ab, then a ±1 or b =±E1. This is probably the definition you were given for prime in your grade school algebra class. It turns out that an integer is prime if and only if it is irreducible. However, the notions of irreducible and prime are not the same in other contexts you will encounter in Math 100/103. Prove Theorem I below. You may use the following fact without proof: for every positive integer n > 2, either n is prime, or there exist positive integers a, b such that ab and 1 < a="" 2="" can="" be="" written="" as="" a="" product="" of="" primes.="" in="" other="" words,="" for="" each="" n="" e="" z,="" with="" n=""> 2, there is r E Z4 and primes P1, P2, . .. , Pr such that п — || Pi = PiP2 Pr. i=1