Linear Combination - Number Theory - Solved Exam, Exams of Number Theory

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Name: SouwnosS MATH 506 Number Theory — Exam II ‘Wednesday March 28, 2012 Check that that you have all three pages - note that page two is on the back of page one 1. (14 points) Let a= 23757137, b = 253913%19, B2 os (a) The prime factorization of ged(a,b) = 2-3: \3 $3 a + (b) ‘The prime factorization of lem(a,b) =_2: 3-3 * BA (c) If 3°|| Ga? + 50? thane = _ S Ss Sy co, Sse f= Tey bb Gels ise = BAW SB | (d) If 3/][100! then f= _ 458. 2. (18 points) (a) Evaluate 7(700)=_ 18 Foo = a.s.3 Tl SPs BaD (b) Evaluate o(700) =_V3b . SC Say a SASS TA) Greed ieee s Cy +) li (c) What prime factorizations are possible for n if r(n) = 6? ne Ot ashe ris b ne ee Pee Pe SD Ten) = Wary Cae ti) HA RA, MSR ne Pe (d) If p and g are distinct primes then r(p’q°) =__|o.__. List the divisors of p®g® (table form is fine), tug Py = Ba =iB 3. (6 points) Give (with justification) a non-trivial factor of 24+ 1. a = (ay : A N+ \ ak a 5 wey ‘r z _ _ v | = a NlLEL s+ a +1) a oe Gnadsy)| = PG SCORE/IS } 1 | |