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

The 4:25 bus to the metro airport has a travel time of 1 hour and 45 minutes. The bus is running late today though. What will be the arrival time to the airport?

Answers

Step-by-step explanation:

what is that ? another drug or other crime gang secret message to each other ?

as you did not specify how much delayed it is, the answer can only be : late.

Help help help pelesss please

Answers

Answer:

its 1

Step-by-step explanation:

it literly says it on the grid and its really simple

least common denominator of both 4/x^2*t and x/t^2

Answers

Step-by-step explanation:

LCD of 4/x²t and x/t²

= LCM(x²t , t²)

=

Use set builder notation to write the set E= {1, 3, 9, 27}

PLEASE NO SPAMMING I NEED AN ANSWER NOW​

Answers

Answer:

E = { x | x = 3ⁿ, n ∈ N, and 0 ≤ n ≤ 3 }

Step-by-step explanation:

Given set:

E = {1, 3, 9, 27}

We observe that:

1 = 3⁰, 3 = 3¹, 9 = 3², 27 = 3³

Each term can be shown as 3ⁿ, where n is natural number between 0 and 3.

Set builder notation:

E = { x | x = 3ⁿ, n ∈ N, and 0 ≤ n ≤ 3 }
E={1,3,9,27}

Here let's observe

1=3^03=3^19=3^227=3^3

So set builder form:-

[tex]\\ \sf\longmapsto E=\{x|x=3^n,n\in N,0\leqslant n\leqslant 3\}[/tex]

Simplify the liner expression
−2/3a+1/8−1/6a−3/4

Answers

This is how far you can simplify it

Does Timmy really brush his teeth before bedtime?
Remember when you were a kid and you were told to brush your teeth before
going to bed? How many of us pretended to brush our teeth thinking that no one
would notice? (It probably took more time to pretend than it would have been to
just brush our teeth in the first place.)
Here we have a null hypothesis, an alternative hypothesis, and the data (evidence).
Null the
Were Hypothesis
They rushed
His Teeth
Teled Not truth
e Tea
The Data
Dry Toothbrush
• What level of significance would you choose, 0.1 or 0.01? Explain.
. Would you reject or not reject the null hypothesis based on the data (a dry
toothbrush)? Explain.
Is there anything that could go wrong with your decision? Explain.
• If something could go wrong, what type of error would that be?

Answers

Answer:

no he doesnt brush his teeth

Step-by-step explanation:

Question 11 (5 points)
22
Find the volume of the solid. Use
7
for II.
5 cm
1 cm
6 cm

Answers

Answer:

Volume of solid = 5 cm *  1 cm  * 6 cm = 30 cm³

Step-by-step explanation:

The formula for volume of solid = Length * Width * Height

How to divide 4 divided by 2/3

Answers

According to your question, (4 divided by 2/3):-

[tex] \frac{4}{ \frac{2}{3} } [/tex]

By reciprocal, it. becomes:

[tex]4 \times \frac{3}{2} [/tex]

As we know that 2×2 is 4, cancel 4. Then we get:-

[tex]2 \times 3[/tex]

[tex]→6[/tex]

Therefore, the answer is 6

What number should be placed in the box to help complete the division calculation? (1 point) Long division setup showing an incomplete calculation. 19 is in the divisor, 8216 is in the dividend, and 4 hundreds and 3 tens is written in the quotient. 7600 is subtracted from 8216 to give 616. An unknown value represented by a box is being subtracted from 616.

Answers

Answer:

The unknown value is being subtracted from 702 is 650

Step-by-step explanation:

Long division setup showing an incomplete calculation

13 is in the divisor

3302 is in the dividend

2 hundreds and 5 tens is written in the quotient

2600 is subtracted from 3302 to give 702

An unknown value represented by a box is being subtracted from 702

We need to find this number

Lets write the steps above

∵ The dividend = 3302

∵ The divisor = 13

- 2 hundreds means 200 and 5 tens means 50

∵ The quotient = 200 + 50 + x

∵ Dividend = divisor × quotient

∴ (13 × 200) + (13 × 50) + (13 × x) = 3302

∵ 13 × 200 = 2600

- Subtract 2600 from the dividend

∴ 3302 - 2600 = 702

∵ 13 × 50 = 650

∴ 702 - 650 = 52 ⇒ 650 is the unknown value

∵ 13 × x = 13x

∵ 52 - 13x = 0

- Add 13 x to both sides

∴ 52 = 13x

- Divide both sides by 13

∴ x = 4

∴ 3302 ÷ 13 = 200 + 50 + 4

∴ 3302 ÷ 13 = 254

∴ From the steps above the missing number subtracted from

  702 is 650

Solve
-16 - 8

A. 24

B. 8

C. -8

D. -24

Answers

Answer:

D. 24

Step-by-step explanation:

This can be rewritten as

-16 + -8

This is equal to -24

Answer:

D= -24

Step-by-step explanation:

coz I used the table method so I also found it's correct

This is due today please helpppp meeee I really need help



Carla and Diane Bought identical suit cases at a different stores. Carla’s suitcase originally cost $85 and was on sale for 20% off the original price which suitcase was a the better deal? What do we know ? List important facts ? What do we want to find out ?? Solve ??? Show ur work? Explain starts ur Asnwer and give details ? And justify your steps ? Someone please help!!!

Answers

Answer:

Carla's suitcase was a better purchase.

Step-by-step explanation:

The complete question is:

Carla and Daniel bought identical suitcases at different stores. Carla's suitcase originally cost $75 and was discounted 19% Daniel's suitcase original cost $85 and was on sale for 20% off the original price. Which suitcase was the cheaper?

Solution:

Cost Price of Carla's suitcase = $75

       Discount% for Carla = 19%

       The amount paid by Carla is,

       

       So, the amount paid by Carla was, $60.75.

Cost Price of Jeana's suitcase = $85

       Discount% for Jeana = 20%

       The amount paid by Jeana is,

       

       So, the amount paid by Jeana was, $68.

Carla's suitcase was a better purchase.

Not sure how to do this.

Answers

Answer:

5ft into 7ft

Step-by-step explanation:

5ft to 7ft I hope I’m right

On a map 1 inch equas 27 miles. How much is 3 inches

Answers

Answer:

3 × 27

=81

mark BRAINLIEST pls

Two customers went to a post office to buy postcards and large envelopes. Each postcard costs the same amount, and each large envelope costs the same amount. The first customer paid $12 for 14 postcards and 5 large envelopes. The second customer paid $24.80 for 10 postcards and 15 envelopes. What was the cost in dollars of each large envelope?
A-$1.42
B- $1.15
C- $0.35
D- $0.63

Answers

The cost in dollars of each large envelope is $1.42.

Two equations can be derived from this question:

14p + 5l = 12 equation 1

10p + 15l = 24.80 equation 2

Where:

p = postcards

l = large envelopes

To determine the cost of large envelopes, multiply equation 1 by 10 and equation 2 by 14.

140p + 50l = 120 equation 3

140p + 210 = 347.20 equation 4

Subtract equation 3 from 4

227.20 = 160l

Divide both sides of the equation by 160

l = 227.20 / 160

l  = $1.42

To learn more, please check: brainly.com/question/23589883

Maria just got a raise at work. She was making $9 per hour, but now she makes 150% of that amount. Explain how you would find the amount of money Maria now makes per hour.

Answers

Answer:

9.00 x 150%= $13.50 an hour

Step-by-step explanation:

maria started to make $9.00 and hour

you multiply 9.00 x 150= $13.50

hope this helps!

Please help me ASAP please please

Answers

Answer:

67

Step-by-step explanation:

CB is 90 degrees, subtract that from 23 degrees, you get 67

Answer:

67 brainlest

Step-by-step explanation:

On a map, the scale is 1 inch = 40 miles. What is the actual distance for 3.25 inches shown on the map?

Answers

Answer:

130 Miles

Step-by-step explanation:

be precise yesterday

Answers

Answer:

is this supposed to be a free point question? I'm confuzzled

right answers only plz and thank you

Answers

Answer:

letter A

Step-by-step explanation:

3x=0

x=0

x-2=0

x=2

2x+7=0

2x=-7

x=-7/2

What is (−47)−(−107)=?
please help

Answers

60

Step-by-step explanation:

(-47)-(-107)

(-47)-1(-107)

-47+107

=60

5) Mike and Mei each improved their yards by planting daylilies and shrubs. They bought their
supplies from the same store. Mike spent $70 on 5 daylilies and 5 shrubs. Mei spent $62 on 1
daylily and 5 shrubs. Find the cost of one daylily and the cost of one shrub.

Answers

1 shrub cost $12
1 daylily cost $2

1 D: $2
5 S: $60
60+2= 62

5*2=10
5*12=60
60+10=70

hope this helps :)

Solve this quadratic equation by factorisation:

2y² + 4y – 30 = 0

my answer is

(2y+6)×(y-5)
2y+6=0 or y-5=0
y=-3 or y=5
is that correct?

Answers

Answer:

Y=5 y=-3

Step-by-step explanation:

Two real solutions:

y =(2+√64)/2=1+4= 5.000

or:

y =(2-√64)/2=1-4= -3.000

Pls help asap the best I can do is 50 points
Answer the ones where the zeros are those are the one that I did bad or left blank

Answers

Step-by-step explanation:

8.a.

a function is something that looks like

y = ax + b or similar.

it describes the way to calculate the result for whatever input value you can come up with.

so,

C = 12×d + 8

you could also say

C(d) = 12×d + 8

as, when you rent, you have to pay these $8 in any case, no matter how long you stay, and you need to pay $12 per day you stay.

funny thing is : you understood this perfectly well when doing b.

c.

the constant rate of change is always the factor of the input variable. here : 12

it is the ratio of (y change) / (x change)

or here (C change / d change).

it indicates that for any increase of d by 1 (staying one day longer) the charges C increase by 12.

9.a.

remember, what we just said about the rate of change ?

it is (result change) / (input change).

between 2010 and 2015

the input changed by +5 (from 2010 to 2015).

the result changed by +6.25 (from 37 to 43.25).

the rate of change is therefore

6.25/5 = 1.25 = 1 1/4 = 5/4

the function is therefore

population in millions = (years since 2010)×5/4 + 37

p = y×5/4 + 37

b.

now we want to find the number of years (y) so that the result of that equation is 50.

50 = y×5/4 + 37

13 = y×5/4

52 = 5y

y = 52/5 = 10.4

so, this will be during the 11th year after 2010 = 2021.

10.

be careful and concentrate.

you see that the sequence goes up 20, when n goes up 5 (and not just 1) ? there was your initial mistake.

a.

again, remember the change rate : it is result change / input change.

so, we have here 20/5 = 4.

so,

C(n) = 4×n + 50

b.

n = 12

C(12) = 4×12 + 50 = 98

c.

now we have C(n), and want to find for which n this is true.

C(n) = 78

78 = 4×n + 50

28 = 4×n

n = 7

after 7 visits the costs are 78.

there are the basic costs of 50 (for n = 0 or 0 visits) and then extra 4 for every visit.

d.

the x-axis is for the number of visits, the y-axis for the costs.

so, the line starts at the point (0, 50), and then passes through the points of the original table (5, 70), (10, 90), (15, 110). and if you want you can calculate the points for higher x (or number of visits).

find the gradient of a line perpendicular to y=-3/4+6

Answers

Answer:

  4/3

Step-by-step explanation:

Assuming your line is intended to be ...

  y = -3/4x +6

The slope is the coefficient of x: -3/4.

The slope of a line perpendicular to it is the opposite of the reciprocal of this, so is ...

  -1/(-3/4) = 4/3

The gradient of the perpendicular line is 4/3.

What is 2346 to the power of 3?

Answers

Answer:

12911717736

Step-by-step explanation:

"236" to the power of 3 is the same thing as multiply 236 by itself 3 times.

236 · 236 · 236 55696 · 236 12911717736

Therefore, the answer is 12911717736.

Answer:

12911717736

Step-by-step explanation:

You're basically multiplying 2346 by itself 3 times which is why you get 12911717736

What is the solution to -4(8- 3x) >6x - 8?
A)x>-4/3
B)x<-4/3
C)x>4
D)x<4

Answers

I think you got your answer

Musa had 3/ 5 ton of grain for all the animals on his large farm. He used 1 /20 ton each day to feed all his animals. How many days was it until Musa needed more grain?

Answers

Answer:

12 days

Step-by-step explanation:

'd' = number of days

1/20d ≤ 3/5

multiply each side by 20 to get:

d ≤ 60/5 or d ≤ 12

Write the equation of the line passing through points (-1,1) and (1,1).

Answers

Answer: x+1

Step-by-step explanation:

Answer:

Try x+1

Step-by-step explanation:

Hope it helps

Is the point (4, -15) on the line y = -3x - 3​

Answers

Answer:

yes

Step-by-step explanation:

since the format of ordered pairs is (x,y) we can plug in the values into the equation to see if they are on the line

y = -3x - 3​

-15 = -3(4)-3........plug in values

-15 = -12 - 3........multiply

-15 = -15..............subtract

-15 = -15..............this is true

Answer:

yes

Step-by-step explanation:

y = -3x-3

Substitute the point into the equation

-15 = -3(4) -3

-15 =-12-3

-15=-15

This is true so the point is on the line

Find the value of x.

Answers

parallel lines rules says that adjesent angles are equal. which will give me 69

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