2. (a) Use a prove by contradiction to show that if a and b are nonzero integers such that a divides b and a + b is odd, then a is odd. (b) Prove that if n is a positive integer, then n² does not equal 2(mod 4).

2. (a) Use A Prove By Contradiction To Show That If A And B Are Nonzero Integers Such That A Divides

Answers

Answer 1

a. Assume a is even, so a = 2k for some integer k. Now let a and b be integers such that a divides b and a + b is odd.

Since a divides b, b = an for integer n, and in turn b = 2nk, which means b is even and hence a + b is also even. But this contradicts our initial assumption, so a must be odd.

b. Let n be even, so that n = 2k for some integer k. Then

n² = (2k)² = 4k²

so that n² ≡ 0 (mod 4).

Now let n be odd, so n = 2k + 1 for integer k. Then

n² = (2k + 1)² = 4k² + 4k + 1

so that n² ≡ 1 (mod 4).

Therefore n² is never congruent to 2 (mod 4).


Related Questions

Solve the Quadratic

Solve for x. Write in simplified radical form.

2x^2 + 3x – 20 = 0

Solution 1:
Solution 2:

Answers

~quadratic equation

======================

2x² + 3x - 20 = 0

= a + b = 3

= ab = 2(-20) = -40

= a = -5

= b = 8

= (2x² - 5x) + (8x - 20)

= x(2x - 5) + 4(2x - 5)

= (2x - 5) (x + 4)

= Solution 1 : x = 5/2 ✔️

Solution 2 : x = -4 ✔️

~ nice to help you ^^

[tex]2x^2 +3x -20 =0\\\\\\\implies x = \dfrac{-3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-20)}}{2 \cdot 2}\\\\\\\implies x = \dfrac{-3 \pm \sqrt{169}}{4}\\\\\\\implies x = \dfrac{-3\pm 13} 4 \\\\\\\text{Hence} \\\\x = \dfrac{-3+ 13}{4} = \dfrac{10} 4 = \dfrac{5}2\\\\\text{Or}\\\\x= \dfrac{-3-13}4 = -\dfrac{16}4 =-4[/tex]

An unopened bag of gummy bears weighs 2 1/3 ounces. Drayven has 3/4 of a bag of gummy bears and Akila has 2/3 of a bag. Who has more? If they put their bags together, would they have more than one bag? Explain your answer.

Answers

Answer:

Drayven has more

Step-by-step explanation:

If you look at the fractions through a drawing you would see that my claim is correct.

How many solutions for
55l=50l+5

Answers

Answer:

Zero

Step-by-step explanation:

The question is impossible to answer. 501+5 does not equal 551, 501+5 equals 506.

2. If (2, 3) is an ordered pair of the
function F(x), which of the following
must be an ordered pair of the inverse
of F(x)?
Answer Choices:
(3, 2)
(-2,-3)
(2,3)
(-3,-2)

Answers

Answer:

Inverse:  (3, 2)

Step-by-step explanation:

Given the point, (2, 3), as an ordered pair of the function, f(x):

Since we know that for any functions, there is a one-to-one correspondence between each domain and range values.  Conversely, each range value also corresponds to one domain value.  

The inverse of a function, [tex]f^{-1}(x)[/tex], also has a one-to-one correspondence. Simply put, we are essentially switching the x and y values to determine the inverse of a function.

Therefore, given the ordered pair from f(x) = (2, 3), then the corresponding coordinates in the inverse function, [tex]f^{-1}(x)[/tex], must be (3, 2).

At the store the ratio of books sold to movies sold was 4:5.For every_books sold there were _ movies sold.

Answers

Answer:

At the store the ratio of books sold to movies sold was 4:5.For every 4 books sold there were 5 movies sold.

Step-by-step explanation:

Suppose you are reviewing your cell phone bill on Friday and you notice you go over your data limit 49% of the time. You
review the weather forecast for tomorrow that says there is a 11% chance of rain. Assuming rain and going over in data are
independent events, what is the probability that it rains and you go over in data tomorrow? Give your answer as an exact
fraction and reduce the fraction as much as possible.

Answers

Probability that it rains and you go over in data tomorrow = 0.0539

Percentage of time it takes to go over your data limit = 49%

Probability that you go over your data limit, P(data) = 49/100

P(data) = 0.49

The chance that it rains tomorrow = 11%

Probability that it rains tomorrow, P(rain) = 11/100

Probability that it rains tomorrow, P(rain) = 0.11

Probability that it rains and you go over your data tomorrow = P(rain) x P(data)

Probability that it rains and you go over in data tomorrow = 0.11 x 0.49

Probability that it rains and you go over in data tomorrow = 0.0539

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dont sound so smart doing it she will know ​

Answers

Jeremy is incorrect because a negative plus a negative will always be a negative.

Natasha earns $425 a week. How many weeks does it take before she has
earned more than $12, 000

Answers

Answer:

29

Step-by-step explanation:

12000/425

passes through (2,-3) with a slope of -9/2

Answers

Answer:

y = 9/2x + 6

Step-by-step explanation:

m= —9/2

y — y1 = m (x — x1)

y + 3 = —9/2 (x — 2)

y = — 9/2x + 6

Felicia bought 20.8 gallons of gasoline. There are approximately 3.8 liters in 1 gallon.

Answers

Answer:

79.04 liters

Step-by-step explanation:

20.8 x 3.8 = 79.04

Express 24 minutes as a fraction of 2'/2 hours. ​

Answers

Step-by-step explanation:

1hr = 60

½ hr = 30

60 + 30 = 90

Find x using the pictures below please. Will mark branliest for first reply.

Answers

Step-by-step explanation:

a) We are going g to use the definition of tangent to find x:

[tex]\tan{18°} = \dfrac{opp}{adj} = \dfrac{15}{x}[/tex]

[tex]\Rightarrow x = \dfrac{15}{\tan{18°}} = 46.2[/tex]

b) To solve this problem, we are going to use the definition of cosine:

[tex]\cos{70°} = \dfrac{adj}{hyp} = \dfrac{x}{46}[/tex]

[tex]\Rightarrow x = 46\cos{70°} = 15.7[/tex]

What would you do if you found out that you wrote a cheque of $78 but it was recorded as $98 in check register?

Answers

Answer:

I will cryy because I only want to write 78 why it recorded as 98

An eight-sided dice numbered from 1 to 8 is rolled 80 times to determine whether it is fair, a If the dice is fair, how many of each number would you expect to get?

Answers

Answer:

10 times for each number

Step-by-step explanation:

:)

Answer:

10

Step-by-step explanation:

if the dice is fair, the sides will land one time every 8, so if its rolled 80 time, and it was equal 10 would be your answer.

help me please !!!!!!

KS3 YEAR 8 maths!


super confused ‍♀️‍♀️

Answers

Answer:

9:29pm

Step-by-step explanation:

How do I do this math !

Answers

Answer:

The equation of a straight line is always

[tex]y=mx+b[/tex]

The constant m always refers to the slope, and the constant b always refers to the y-intercept.

Simply substitute the numbers in:

[tex]y=-\frac{1}{2}x+3[/tex]

There's your equation!

Write an algebraic expression for the verbal expression: "7 times the sum of a squared and b minus 4 times the sum of a squared and b".​

Answers

7x(b2-4xb2) the b2 meaning b squared

an oven has a maximum temperature of 220°c.what will it be in 2 days time​

Answers

Answer:

Well I assume 220ºC.

Step-by-step explanation:

That is the MAXIMUM temperature of the oven

A waiter carried 6 baskets with 5 dinner rolls in each
basket. How many dinner rolls did he carry? Show
your work.


Answers

Answer:

30 dinner rolls

Step-by-step explanation:

There are 5 dinner rolls per basket and there are 6 baskets

So there are 6 groups of 5:

5 + 5 + 5 + 5 + 5 + 5

In other words, multiply 5 and 6

5 dinner rolls x 6 = 30 dinner rolls

5x6=30

Each basket has five rolls so you have to do 5+5+5+5+5+5 which equals 30

Which number should be multiplied by 45 so that it will have 3 nct prime factors?​

Answers

Answer:

2

Step-by-step explanation:

These points are linear.
Find the slope.
M=?

Answers

Answer:

Step-by-step explanation:

(5, 0) (6, 8)

(8 - 0)/(6 - 5) = 8/1 = 8

Question
Given that the market basket for gardening was $\$131.17$ last year and $\$131.96$ this year, what was the inflation rate?
(Round your answer to the nearest hundredth of a percent.)

Answers

Answer:

0.006%

Step-by-step explanation:

minused old price from the new one and then divided the answer by the old price

50° x
how to find value of unknown angle​

Answers

Answer:

a) 50

b) 86

c) x = 130      y = 50

Step-by-step explanation:

Opposite angles that share the same lines are always the same.

Write the equation of the line in slope intercept form

Slope =−1/5 Y-Intercept =−3

Answers

Answer:

y = mx + c ......( general form)

y = -1/5X - 3

The percent of Ivy Tech students receiving financial aid in 2018 was 82%, this was a 9.8% relative increase from 2017. What would the percent of students receiving aid in 2017 have been?

Answers

In 2017 it would have been 72.2%

Question 1 of 10
f (2) = y + 10. Find the inverse of f(x).
O A f-'(x) = (x + 10)
O B. f-1(x) = 73 + 10
O C. f-1(x) = (x – 10)3
O D. f-'(x) = x3 – 10

Answers

Option D

step by step explanation:

[tex] \sf f(x) = \sqrt[3]{x + 10} \\ \sf y = \sqrt[3]{x + 10} \\ \sf y = \sqrt[3]{y + 10} \\ \sf y = {x}^{3} - 10 \\ \sf {f}^{ - 1} (x) = {x}^{3} - 10[/tex]

The inverse of the given function is represented as f' ( x ) = x³ - 10

What is Function Rule?

A function is a mathematical object that takes an input (or several inputs) and produces an output. The rule of a function describes how the input is transformed to produce the output. It can be written using mathematical notation, words, or a combination of both.

Functions are used in many areas of mathematics, science, engineering, and other fields to model relationships between variables, make predictions, and solve problems.

Given data ,

Let the function be represented as A

Now , the value of A is

Now , the function is depicted as f ( x ) = ∛( x + 10 )    

Simplifying results in

The function's inverse is obtained by swapping the x and y variables.

So ,

y = ∛( x + 10 )

x = ∛( y + 10 )

On solving for y , we get

y = x³ - 10

And , the inverse is f' ( x ) = x³ - 10

The function is therefore f' (x) = x³ - 10.

Go here to read more about the function rule.

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#SPJ7

Crafts Unlimited sells ribbon by the foot. Serena needs ribbon

for an art project. The cost of the ribbon is 5 feet for $3.00.

Feet of Ribbon 1 5 10 15

Cost (in dollars) $3.00

a. Complete this ratio chart based on the information provided.

b. How many feet of ribbon can Serena buy for $12.00?

c. If Serena needs 17 feet of ribbon, how much will she pay?

How do you know?

Answers

The completed ratio chart is provided below:

Feet of Ribbon    1             5   10    15

Cost                     $0.60   $3  $6   $9

The amount Serena would get if she has $12 is 20 yards.

The amount Serena would pay if she wants 17 feet of the ribbon is $10.20.

The first step is to determine the unit cost of a ribbon.

Unit cost of a ribbon = $3 / 5 = $0.60

The cost of 10 feet of the ribbon = $0.60 x 10 = $6

The cost of 15 feet of the ribbon = $0.60 x 15 = $9

The feet of ribbon Serena can buy if she has $12 = $12 / $0.60 = 20 yards.

The cost of 17 feet of the ribbon = $0.60 x 17 = $10.20

To learn more, please check: https://brainly.com/question/12528427

There are m marbles in a bag. Write an expression for the number of marbles in the bag when 4 marbles are taken away​

Answers

Answer:

m-4

Step-by-step explanation:

4 marbles taken away = -4 marbles

Need all of these answered please

Answers

alternate entireror angles are congurent by this rule you can find your answer by solving the equation

What is the temperature change between the greatest value and the least value?
-25°F, 5°F, 60°F, -45°F, -10°F, 90°F
WILL BR MARKED BEST

Answers

The temperature change between the highest and lowest.
Highest- 90 degrees
Lowest- -45 degrees

-45 to 0 = 45 units
0 to 90 = 90 units
45+90 units = 130

ANSWER 130
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