465eps01 - MATH 465 NUMBER THEORY, SPRING TERM 2009,...

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Unformatted text preview: MATH 465 NUMBER THEORY, SPRING TERM 2009, PRACTICE EXAM 1, MODEL SOLUTIONS 1. Suppose that l,m,n ∈ N . Prove that ( lm,ln ) = l ( m,n ). Suppose that m , n , l have the canonical decompositions m = p α 1 1 ...p α s s , n = p β 1 1 ...p β s s , l = p γ 1 1 ...p γ s s where the α j , β j , γ j are non–negative integers. Then ( lm,ln ) = Q j p min( γ j + α j ,γ j + β j ) j and l ( m,n ) = Q j p γ j +min( α j ,β j ) j and the result fol- lows on observing that min( γ + α,γ + β ) = γ + min( α,β ). 2. (i) Show that if ( l, 6) = 1, then l ≡ ± 1 (mod 6). (ii) Show that if l ≡ m ≡ 1 (mod 6), then lm ≡ 1 (mod 6). (iii) Show that if lm ≡ - 1 (mod 6), then either l ≡ - 1 (mod 6) or m ≡ - 1 (mod 6). (iv) Show that if n ∈ N and n ≡ - 1 (mod 6), then there is a prime number p such that p | n and p ≡ - 1 (mod 6). (v) Show that there are infinitely many primes of the form 6 k- 1....
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This note was uploaded on 10/03/2011 for the course CMPSC 465 taught by Professor Burchardcharles during the Spring '08 term at Pennsylvania State University, University Park.

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