Let Zn* = {x| 1≤x ≤ n-1, where GCD(x, n)=1 }. What are the elements of Z16*?
Elements of Z16* include
{1,3,5,7,9,11,13,15}.
They can be found by writing a program that loops from 1 to n-1 and calculates gcd and sees if it is equal to 1 or not.
Pseudo code:
result_set = []
for x in 1..(n-1) {
if(gcd(n,i)==1) result_set.append(i)
}
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Let Zn* = {x| 1≤x ≤ n-1, where GCD(x, n)=1 }. What are the elements of...
10.) Consider (Zn; n), the set Zn with mod n
multiplication.
i. Argue that if neither a nor b has any common divisors greater
than 1
with n then neither does ab. [Equivalently gcd(a; n) = 1,
etc.]
ii. Argue that if a does not have any common divisors greater than
1
with n, then [a]n has a multiplicative inverse in Zn.
iii. Argue that (i) and (ii) imply that the set of elements
f[a]n 2 Znjgcd(a; n) = 1g...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
Let a and be be in . Show
the following. If gcd(a,b)=1, then for every n in there
exist x and y in such
that n=ax+by.
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Consider N and the set S={x∈{0,...N-1}:gcd(x,N)=1} where k=|S| For a∈S, we define T={ax(modN):x∈S}. what is |T|? Answer may include N and k.
Let p and n be integers. Prove that, if p is prime, then gcd(p, n) = p or gcd(p, n) = 1. . . (i.) Using proof by contrapositive (ii.) Using proof by contradiction
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
Let A be a set with m elements and B a set of n elements, where m, n are positive integers. Find the number of one-to-one functions from A to B.
C1= 5
C2= 6
C3= 10
GCD --> Greater Common Divisor
B1 a. Let x := 3C1 + 1 and let y := 5C2 + 1. Use the Euclidean algorithm to determine the GCD (x, y), and we denote this integer by g. b. Reverse the steps in this algorithm to find integers a and b with ax + by = g. c. Use this to find the inverse of x modulo y. If the inverse doesn't exist why not?...
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