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

This is the Solved Exam of Number Theory which includes Last Complete Solution, Prime, Integer Relatively, Root Modulo, Distinct Primes, Primitive Root Modulo, Discrete Logarithms, Incongruent etc. Key important points are: Prime Factorization, Remainder, Induction, Non Trivial Factor, Right Angled Triangles, Integer Sides, Given Length, Multiple, Divided, Chinese Remainder Theorem

Typology: Exams

2012/2013

Uploaded on 02/18/2013

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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