a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo...

a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo n is well defined as a residue class in Zn, and depends only on the residue class a in Zn.

b) Let Z × n ⊆ Zn be the set of invertible residue classes modulo n. Prove that Z × n forms a group under multiplication. Is this group a subgroup of Zn?

c) List the elements of Z × 9 . How many are there? For each residue class u ∈ Z9, compute the elements of the sequence u, u2 , u3 , u4 , . . . until the pattern is clear. Determine the length of each repeating cycle. Is Z × 9 a cyclic group?

Expert Solution

Colby Messinger answered 2 years ago

4. Prove that for any n∈Z+, An ≤Sn.

let n belongs to N and let a, b belong to Z. prove that a is...

let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.

1. Determine an inverse of a modulo m for a = 6 and m = 11. This...

1. Determine an inverse of a modulo m for a = 6 and m = 11. This is equivalent to answering the question “_______ is the unique inverse of 6 (mod 11) that is non-negative and < 11.” Show your work following the steps. Beside the inverse you identified in part a), identify two other inverses of 6 (mod 11). Hint: All of these inverses are congruent to each other mod 11. Although the congruence can be solved using any of the inverses...

Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2...

Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2 + 2.

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈...

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈ Zn | there is a [b] ∈ Zn with [a] ·n [b] = [1]} (a) Show that (Un, ·n ) is a group. (b) Write down the Cayley table for U5. Hint: |U5| = 4. (c) Write down the Cayley table for U12. Hint: |U12| = 4.

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo...

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo p2 . (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.) (b) Prove that for any n, there is a primitive root modulo pn. (c) Explicitly find a primitive root modulo 125. Please do all parts. Thank you in advance

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a) Suppose that a ∈ Z is a unit modulo n. Prove that its inverse modulo...

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4. Prove that for any n∈Z+, An ≤Sn.

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