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).
Answers
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
Not sure how to do this.
Answers
Answer:
5ft into 7ft
Step-by-step explanation:
What is (−47)−(−107)=?
please help
Answers
60
Step-by-step explanation:
(-47)-(-107)
(-47)-1(-107)
-47+107
=60
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
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
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
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!
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
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
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 12911717736Therefore, 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
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.
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
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:
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
On a map 1 inch equas 27 miles. How much is 3 inches
Answers
Answer:
3 × 27
=81
mark BRAINLIEST pls
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 = 3³Each 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 }Here let's observe
1=3^03=3^19=3^227=3^3So set builder form:-
[tex]\\ \sf\longmapsto E=\{x|x=3^n,n\in N,0\leqslant n\leqslant 3\}[/tex]
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.
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:
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.
Help help help pelesss please
Answers
Answer:
its 1
Step-by-step explanation:
it literly says it on the grid and its really simple
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
Simplify the liner expression
−2/3a+1/8−1/6a−3/4
Answers
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 daylily cost $2
1 D: $2
5 S: $60
60+2= 62
5*2=10
5*12=60
60+10=70
hope this helps :)
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
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
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:
be precise yesterday
Answers
Answer:
is this supposed to be a free point question? I'm confuzzled
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)
= x²t²
Find the value of x.
Answers
parallel lines rules says that adjesent angles are equal. which will give me 69
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).