CSci 5471: Modern Cryptography, Spring 2010 Homework #1
Yongdae Kim
–
Due: Feb. 2 9:00 AM
–
Type up your solution (either text or pdf) and send it to me and TA by email. (
kyd(atmark)cs.umn.edu,
yongdaek(atmark)gmail.com,hkang(atmark)cs.umn.edu
)
–
Show all steps. Please ask any questions, if you find any problems.
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Total 100 points + 20 points extra credit
1.
Discrete Math and Number Theory
(a) (5 pts) Prove or disprove:
n
3
+ (
n
+ 1)
3
+ (
n
+ 2)
3
is always mutiple of
9
. (Hint) Proof by cases
(b) (5 pts) Show that if
2
n

1
is a prime, then
n
is a prime. (Hint) Proof by contradiction.
(c) (5 pts) Show that for all integers
n
,
4
6 
n
2
+ 2
.
(d) (10 points) Let
n
be a positive integer. The Euler function
φ
(
n
)
is defined as the number of nonnegative
integers
k
less than
n
which are relatively prime to
n
. Prove that
i. (3 points)
φ
(
p
) =
p

1
for all prime
p
.
ii. (7 points)
φ
(
p
c
) =
p
c

1
(
p

1)
for all prime
p
and all positive integer
c
.
(e) (10 pts) Show that in any set of
n
+ 1
positive integers not exceeding
2
n
, there must be two integers that are
relatively prime. (Hint) Use Pigeonhole Principles.
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 Spring '08
 Staff
 Number Theory, pts, Prime number, bins, Yongdae Kim

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