Prime Factorization - Number Theory - Solved Exam, Exams of Number Theory

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Name: @owrioss, MATH 506 Number Theory —- Final Exam Monday May 7, 2012 Check that you have all four pages. Show all your work. Assume multiplicativity whore appropriate. 3 1. (4 points) If @ = 2413719, b = 255713 then the prime factorization of (a?, 0) = 2B 2 ee Fei ea we 2. (6 points) (a) Evaluate: (3000) = Acs. 25°) = SG). BA) SS) = Bes (b) The remainder when 114 is divided by 3000 is Jal. Osx Satass Treeuer 2NoL % Boe 2 2a ebizoss> Ws Gey os Few = at Cad sno ) PO 1 Genk Beco) 3. (6 points) Use induction te prove that 2” | (Qn)! Gane Hag: S| aleaa aanks, . - Xe sora MG PN Tadudiie Sap > Assi a’ \ (ax)! ~ Gis ee = Aer ans a | Cacesy)! te (aces)! =a ™ Seen (aces)! = uaryt =(arvsrd fam las)! = & Cer dared oho eat 1 hes mle Leeann e NN. Wee a” \ Quen)! wad TH vast Ratios Fen Fe Pande & Wake Sinn \+ . ew SB $3 wu “ Aey = (ya = (at ola wae a & 5, (12 points) Find all right-angled triangles with coprime integer sides and base of_ given. length: (i) 28 4, (4 points) A non-trivial factor of 2°? +1 is RRO, QS wes, & =x Brn ON=u 2 iN Seer ss Rot . wenn Gage) = laa, 9S, 9) eee = (2%, 45,82) | a ys ee Be F try (ii} 15. Loe gt | = = AG = we IS 2D ves, Nel 2 ow ’ 2Wi7 is wa) Co®) SY > s, ey) Po scone /az