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Hello,
Looking for a competent tutor that is skilled in number theory. I have been working on an assignment and need an alternative to referring to checking my work. I completed the first questions from the book and double-checked those, so you would only be responsible for questions A through the bonus.Please review the attached and do not accept if you are not competent to return the solutions in time.
Homework Assignment 2
Due: Wednesday, February 9
Do problems 12, 16, 18 on pages 47-48 from the book, 2 points each.
A. (2 points)
Let f (n) = 2n + 1 for all n, and let g(n) = n for all n.
X
n
Prove that
f (d)g( ) = 2nd(n) + s(n).
d
d|n
B. (2 points) Let SF (n) be the squarefree function of n. X
Hence, SF (1) =
a1 a2
ar
?2 (d)f(d) =
1 and SF (p1 p2 …pr ) = p1 p2 …pr . Prove the following:
d|n
SF (n).
C. (1 point) Use the fact that 47 = 3 ? 24 – 1 and 35 = 23 mod 47 to show
that 47 | (223 – 1).
D. (3 points) Prove the following:
i) If fi (x) = O(g(x)) for each 1 = i = k then
k
X
fi (x) = O(g(x)).
i=1
ii) If fi (x) = O(gi (x)) and gi (x) = g(x) for each 1 = i = k, then
k
X
fi (x) = O(g(x)).
i=1
and
a is any small, positive real number, then
ZO(g(x))
Z x iii) If f (x) =
x
f (t) dt = O
g(t) dt .
a
a
1-Point Bonus:
Let c be any real number. Show that if f is multiplicative such
X that
r
r-1
r
f (1) = 1 and f (p ) = c(c+1)
for all prime powers p , then F (n) =
f (d)
d|n
is completely multiplicative.
1
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