Question

(a) Give the order of each element of {1,2,3,...,10} modulo 11.

(b) Find all possible products of an element of order 2 with an element of order 5 and show that they give primitive elements modulo 11.

Answer #1

(a) Give the order of each element of {1,2,3,...,10} modulo
11.
(b) Find all possible products of an element of order 2 with an
element of order 5 and show that they give primitive elements
modulo 11.

(i) Verify that 2 is a primitive root modulo 29.
(ii) Find all the primitive roots modulo 29. Explain how you
know you have found them all.
(iii) Find all the incongruent solutions to x6 ≡
5(mod 29).

(a) If a is an integer that is not divisible by 23, what are the
possible values of ord23(a)?
(b) Use part (a) to help show that 5 is a primitive root modulo
23.
(c) Show that 2 is NOT a primitive root modulo 23, by using
part
(b) to help find ord23(2). [Hint: Write 2 as a power of 5 (mod
23).] (d) Use part (b) to help find four more primitive roots
modulo 23

Find the order of the given element 11 in U48
Please give me details..

Let A,BA,B, and CC be sets such that |A|=11|A|=11, |B|=7|B|=7
and |C|=10|C|=10. For each element (x,y)∈A×A(x,y)∈A×A, we associate
with it a one-to-one function f(x,y):B→Cf(x,y):B→C. Prove that
there will be two distinct elements of A×AA×A whose associated
functions have the same range.

Compute the order of each element in the Quaternions, Q8. Show
all work

For Boolean variables A, B and C, list all the possible
products. (Note that order of literals doesn’t matter. For example,
B A is considered to be the same product as A B, and the same is
true for ABC and BCA.)

Find ord5(2), ord7(2),
ord11(2), and ord11(3). Where
ordn(x) represents the order of x modulo n. Do these by
hand and show all work.

2. Let a and b be elements of a group, G, whose identity element
is denoted by e. Prove that ab and ba have the same order. Show all
steps of proof.

6. What is the orbit of 2 in the group Z_7 under multiplication
modulo 7? Is 2 a generator?
7. What is the residue of 101101 modulo 1101 using these as
representations of polynomials with binary coefficients?
8. List all irreducible polynomials with binary coefficients of
degree 4 or less. (Hint: produce a times table that shows the
minimum number of products needed.) Show these as binary numbers
(omitting the indeterminant) and as decimal numbers (interpreting
the binary number into...

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