3) State whether or not each statement is true or false: a) If f : N → Q is such that Σan f(d) = 1/n, then ƒ(6) = 1/3. b) If n is a positive integer and o(n) denotes the sum of its positive divi- sors and u is the Möbius function, then o(1)μ(12)+o(2)μµ(6)+o(3)µ(4)+o(4)μ(3)+o(6)µ(2)+o(12)µ(1) = 12. c) For all odd primes p and and all pairs of integers a, b such that the congruence x² = ab mod p has a solution, the congruences x² = a mod р and x² = b mod p also have solutions. d) If a is a primitive root in Z257, then every odd power of a is again a primitive root.
3) State whether or not each statement is true or false: a) If f : N → Q is such that Σan f(d) = 1/n, then ƒ(6) = 1/3. b) If n is a positive integer and o(n) denotes the sum of its positive divi- sors and u is the Möbius function, then o(1)μ(12)+o(2)μµ(6)+o(3)µ(4)+o(4)μ(3)+o(6)µ(2)+o(12)µ(1) = 12. c) For all odd primes p and and all pairs of integers a, b such that the congruence x² = ab mod p has a solution, the congruences x² = a mod р and x² = b mod p also have solutions. d) If a is a primitive root in Z257, then every odd power of a is again a primitive root.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3) State whether or not each statement is true or false:
a)
If f : N → Q is such that Σan f(d) = 1/n, then ƒ(6) = 1/3.
b) If n is a positive integer and o(n) denotes the sum of its positive divi-
sors and is the Möbius function, then
μ
o(1)μ(12)+o(2)µ(6)+o(3)µ(4)+o(4)μ(3)+o(6)µ(2)+o(12)µ(1) = 12.
c) For all odd primes p and and all pairs of integers a, b such that the
congruence
x² = ab mod p
has a solution, the congruences x² = a mod p and x² = b mod p also
have solutions.
d) If a is a primitive root in Z257, then every odd power of a is again a
primitive root.
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