Question

Number Theory Course , I need a full explained answer for those proofs please

1. Prove that for every integer x, x + 4 is odd if and only if x
+ 7 is even.

2. Prove that for every integer x, if x is odd then there exists an
integer y such that x^2 = 8y + 1.

Answer #1

If n is an odd integer, prove that 12 divides
n2+(n+2)2+(n+4)2+1. Please provide
full solution!

PLEASE DON'T USE ANY OTHER PROOFS FROM POOFS BY CASES! USE
DEFINITION OF ABSOLUTE VALUE!
And I was thinking of doing 4 cases:
1. x greater than or equal to 0, and y greater than
or equal to 0
2. x greater than or equal to 0, y less than 0
3. x less than 0, y greater than or equal to 0
4. x less than 0 , and y is less than 0.
Proofs by cases - absolute value.
Prove...

1. Let x be a real number, and x > 1. Prove 1 < sqrt(x)
and sqrt(x) < x.
2. If x is an integer divisible by 4, and y is an integer that
is not, prove x + y is not divisible by 4.

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

Please I need the full answer in less than 1 hour.
In a faculty of dentistry, the average durability of dental
fillings made using
a particular medication by the assistant as a dissertation. The n =
25 dental fillings made
using the medication are selected randomly and the sample mean and
the sample standard
deviation are calculated as; x=7 year, S= 1.57 year.
A-For the population mean μ, obtain the two sided confidence
interval for 1- α = 0.95. Explain...

please answer all of them; only need final answer
h. The length of the vector c = (2, -3, 1) is?
i. If |r| = 4, then the value of |-5r| is?
j. The cross product of vectors a = (-1, 4, 6) and b = (3, 0, 2)
is?
k. The vectors v = (3, 4) is projected onto the x-axis The
length of the projection is?
l. Rounded to the nearest integer, a two-dimensional vector with
length 13...

Please, I need full explanation and correct answer.
9) In a country with a working-age population of 22 million, 16
million are employed, 2 million are unemployed, and 1 million of
the employed are working part-time, half of whom wish to work
full-time. If 500,000 of those unemployed are cyclically
unemployed, what is the natural unemployment rate? 9)
_______
A) 5.6 percent
B) 9.4 percent
C) 8.3 percent
D) 11.1 percent
E) 1.5 percent

Please I need answer for This question I have assignment
for this subjects . Please be brief and to the point . ,but It is
enough to give the name of the authors Only legible answers will be
considered.Thanks in advance/Ha
1-Theory:
a-Discrimination definition.
b-Statistical discrimination.
c-Preference-based discrimination.
d-Attention discrimination.
e-Explicit Discrimination & Implicit Discrimination.
f-Hypothesis formulation

Please answer Problems 1 and 2 thoroughly.
Problem 1: Let X be a set. Define a partial ordering ≤ on P(X)
by A ≤ B if and only if A ⊆ B. We stated the following two facts in
class. In this exercise you are asked to give a formal proof of
each:
(a) (1 point) If A, B ∈ P(X), then sup{A, B} exists, and sup{A,
B} = A ∪ B.
(b) (1 point) If A, B ∈ P(X),...

Can you please answer number 6! Its the only one i need
thanks!!!
The actual values for 12 periods (shown in order) are:
(1) 45 (2) 52 (3) 48 (4) 59 (5)
55 (6) 55 (7) 64 (8)
58 (9) 73 (10) 66 (11)
69 (12) 74
Using a 5 period simple moving average, the forecast for period
13 will be:
QUESTION 2
Using the 4 period weighted moving average, the forecast for
period 13 will be:
QUESTION 3
With exponential smoothing, the forecast for period 13 will
be:...

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