Question

Prove that if x ∈ Zn − {0} and x has no common divisor with n greater than 1, then x has a multiplicative inverse in (Zn − {0}, ·n).

State the theorem about Euler’s φ function and show why this fact implies it.

Answer #1

Prove the following theorem about rational numbers:
If [(x, y)] ≠ [(0, 1)] then [(x, y)] has a multiplicative
inverse

(2) Letn∈Z+ withn>1. Provethatif[a]n
isaunitinZn,thenforeach[b]n ∈Zn,theequation[a]n⊙x=[b]n has a unique
solution x ∈ Zn.
Note: You must find a solution to the equation and show that
this solution is unique.
(3) Let n ∈ Z+ with n > 1, and let [a]n, [b]n ∈ Zn with
[a]n ̸= [0]n. Prove that, if the equation [a]n ⊙ x = [b]n has no
solution x ∈ Zn, then [a]n must be a zero divisor.

Prove Euler’s theorem: if n and a are positive integers with
gcd(a,n)=1, then aφ(n)≡1 modn, where φ(n) is the Euler’s function
of n.

For any integer n>1, prove that Zn[x]/<x> is isomorphic
to Zn. Please explain best way possible and use First Isomorphism
Theorem for rings.

Use the fact that <[1]>=<[a]>=Zn to prove that the
number of elements of order n in Zn is exactly the Euler phi
function of n.

The greatest common divisor c, of a and b, denoted as c = gcd(a,
b), is the largest number that divides both a and b. One way to
write c is as a linear combination of a and b. Then c is the
smallest natural number such that c = ax+by for x, y ∈ N. We say
that a and b are relatively prime iff gcd(a, b) = 1. Prove that a
and n are relatively prime if and...

Let A ={1-1/n | n is a natural number}
Prove that 0 is a lower bound and 1 is an upper
bound: Start by taking x in A. Then x = 1-1/n
for some natural number n. Starting from the fact that 0 <
1/n < 1 do some algebra and arithmetic to get to 0 < 1-1/n
<1.
Prove that lub(A) = 1: Suppose that
r is another upper bound. Then wts that r<= 1.
Suppose not. Then r<1. So 1-r>0....

Prove the following statement by mathematical induction. For
every integer n ≥ 0, 2n <(n + 2)!
Proof (by mathematical induction): Let P(n) be the inequality 2n
< (n + 2)!.
We will show that P(n) is true for every integer n ≥ 0. Show
that P(0) is true: Before simplifying, the left-hand side of P(0)
is _______ and the right-hand side is ______ . The fact that the
statement is true can be deduced from that fact that 20...

Prove that, for x ∈ C, when |x| < 1, lim_n→∞ |x_n| = 0.
Note:
To prove this, show that an = xn is monotone decreasing and
bounded from below. Apply the Monotone sequence theorem. Then, use
the algebra of limits, say limn→∞ |xn| = A, to prove that A =
0.

If g(x) = x + x^2, prove that g(n) − 4g(m) = 0 has
no solutions for positive integers m and n.

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