Homework for Section 9.2 and 9.3
Dominique Dec
March 29, 2015
Section 9.2
5 Find a complete set of incongruent primitive roots of 13.
- There will be (13) = (12) = 4 incongruent roots. We must rst show that ord13 2 = 12 and that
it must be a divisor of 12
Homework for Section 8.2, 8.3, and 8.4
Dominique Dec
April 12, 2015
Section 8.2
7 Will be attached as an image.
13 Using the digraphic cipher that sends plaintext to ciphertext with C1 3P1 + 10P2 (mod 26), C2
9P1 + 7P2 to encrypt the message: BEWARE OF T
Homework Sections 7.1 and 7.2
Dominique Dec
March 8, 2015
Section 7.1
1 Determine whether each of the following arithmetic functions is completely multiplicative. Prove your
answers.
a f (n) = 0, f (m) = 0 so f (mn) = 0 0 = 0 for all f f (n) = n!; f (4) =
Sections 4.6, 5.2, and 5.5 Homework
Dominique Dec
February 20, 2015
Section 4.6
1 Use the Pollard rho method with x0 = 2 and f (x) = x2 + 1 to nd the prime factorization of each of the
following integers.
d n = 8131, x0 = 2, f (x) = x2 + 1
(x2 x1 , 8131)
Sections 4.2, 4.3 Homework
Dominique Dec
February 15, 2015
Section 4.2
2 Find all solutions of each of the following linear congruences.
a 3x 2 (mod 7) (3, 7) = 1 & 1 | 2, so there is one solution
3x 7y
=
2
7
=
3(2) + 1
3
=
1(3)
1
=
732
2
=
7234
x0 = 4, y
Sections 3.5, 3.6, and 4.1 Homework
Dominique Dec
February 8, 2015
Section 3.5
10 Show that if a, b Z+ and a3 | b2 , then a | b.
Proof. If a3 | b2 then b2 = ka3 when k Z. Also if a | b then b = la for some l Z. From here we can
manipulate the two equation
Homework for 3.3, 3.4, 3.5
Dominique Dec
January 31, 2015
Section 3.3
3 Let a be a postive integer. What is the gcd of a and 2a?
(a, 2a) = a
1, a
1, a , 2, 2a
a :
2a :
4 Let a be a positve integer. What is the gcd of a and a2 ?
(a, a2 ) = a
a
2
a
1, a
: 1
Sections 1.3 and 1.5
Dominique Dec
January 15, 2015
1.3 Homework
2 Conjecture a formula for the sum of the rst n even postive integers. Prove by using mathematical
induction.
S(n) = n 2k
k=1
=
2(1) + 2(2) + 2(3) + . . . + 2(n) = n(n + 1)
1 2k
k=1
n+1 2k
k