Question

Exercise 6.4. We say that a number x divides another number z if there exists an integer k such that xk = z. Prove the following statement. For all natural numbers n, the polynomial x − y divides the polynomial x n − y n.

Answer #1

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

1. Let A ⊆ R and p ∈ R. We say that A is bounded away from p if
there is some c ∈ R+ such that |x − p| ≥ c for all x ∈ A. Prove
that A is bounded away from p if and only if p not equal to A and
the set n { 1 / |x−p| : x ∈ A} is bounded.
2. (a) Let n ∈ natural number(N) , and suppose that k...

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...

In number theory, Wilson’s theorem states that a natural number
n > 1 is prime
if and only if (n − 1)! ≡ −1 (mod n).
(a) Check that 5 is a prime number using Wilson’s theorem.
(b) Let n and m be natural numbers such that m divides n. Prove the
following statement
“For any integer a, if a ≡ −1 (mod n), then a ≡ −1 (mod m).”
You may need this fact in doing (c).
(c) The...

Define the relation τ on Z by aτ b if and only if there exists x
∈ {1,4,16} such that
ax ≡ b (mod 63).
(a) Prove that τ is an equivalence relation.
(b) Prove that there exists an integer n with 1 ≤ n ≤ 62 such
that the equivalence class of n is{m ∈ Z | m ≡ n (mod 63)}.

(a) Prove that if y = 4k for k ≥ 1, then there exists a
primitive Pythagorean triple (x, y, z) containing y.
(b) Prove that if x = 2k+1 is any odd positive integer greater
than 1, then there exists a primitive Pythagorean triple (x, y, z)
containing x.
(c) Find primitive Pythagorean triples (x, y, z) for each of z =
25, 65, 85. Then show that there is no primitive Pythagorean triple
(x, y, z) with z...

1. Suppose we have the following relation defined on Z. We say
that a ∼ b iff 2 divides a + b. (a) Prove that the relation ∼
defines an equivalence relation on Z. (b) Describe the equivalence
classes under ∼ .
2. Suppose we have the following relation defined on Z. We say
that a ' b iff 3 divides a + b. It is simple to show that that the
relation ' is symmetric, so we will leave...

When we say Prove or disprove the
following statements, “Prove” means you show the
statement is true proving the correct statement using at most 3
lines or referring to a textbook theorem.
“Disprove” means you show a statement is wrong by
giving a counterexample why that is not true).
Are the following statements true or not? Prove or disprove
these one by one. Show how the random variable X looks in each
case.
(a) E[X] < 0 for some random...

python
If a number num1 divides another number num2
evenly then num1 is a divisor of num2. For
example, 2 is a divisor of 2, 4, 6, 8, but 2 is not a divisor of 1,
3, 5, 7, 9, 11, 13.
Write a function named count_divisors(m,n) that works as
follows.
Input: the function takes two integer arguments m and n
Process: the function asks the user to enter numbers (positive
or negative), and counts the total number of inputs...

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 11 minutes ago

asked 18 minutes ago

asked 22 minutes ago

asked 25 minutes ago

asked 1 hour ago

asked 3 hours ago

asked 3 hours ago

asked 4 hours ago

asked 5 hours ago

asked 5 hours ago

asked 6 hours ago

asked 6 hours ago