Chinese Remainder Theorem - Number Theory - Exam, Exams of Number Theory

Download Chinese Remainder Theorem - Number Theory - Exam and more Exams Number Theory in PDF only on Docsity! pg score/30 Name: MATH 506 Number Theory – Exam III Wednesday April 25, 2012 Check that that you have all three pages. You may assume multiplicativity where appropriate. τ(n) = the number of positive divisors of n, σ(n) = the sum of the positive divisors of n, φ(n) = the Euler phi-function, µ(n) = the Möbius function, ω(n) = number prime divisors of n. 1. (6 points) (a) Evaluate φ(3000) = (b) Evaluate µ(30) = 2. (12 points) Use the Chinese Remainder Theorem to solve the system of simultaneous congruences: x ≡ 1 (mod 5) x ≡ 3 (mod 6) x ≡ 6 (mod 7) 3. (12 points) (a) Find (132, 385) and write this g.c.d. as a linear combination of the two numbers. (b) Solve the following congruence or say why no solutions exist: 132x ≡ 297 (mod 385) 1 pg score/29 4. (15 points) Define F (n) = ∑ d|n µ(d)d. (a) A function F (n) is multiplicative if . (b) Prove that the above F (n) is multiplicative. You may assume that µ(n) is multiplicative. (c) If p is a prime then F (pk) = (d) Evaluate F (3500) = 5. (14 points) Suppose that f(n) is the multiplicative function satisfying µ(n)τ(n) = ∑ d|n f(d). (a) From the Möbius Inversion Formula f(n) = ∑ d|n . (b) If p is a prime then f(p) = , f(p2) = and f(p3) = . (c) Evaluate f(300) = 2