The that a must have a Prime divisor of the form p = 4m 3 for some m ∈ z, as required.
To prove that a has a prime divisor p of the form p = 4m 3 for some m ∈ z, we need to use a proof by contradiction. Assume that a does not have a prime divisor of the form p = 4m 3 for any m ∈ z. This means that all prime divisors of a must be of the form p = 4m 1 or p = 2.
First, let's consider the case where all prime divisors of a are of the form p = 4m 1. Since a = 4n 3, we know that it is odd and not divisible by 2. Therefore, all its prime divisors must also be odd, which means they can be expressed as p = 4m 1. However, we can easily see that the product of any number of primes of the form 4m 1 is also of the form 4m 1. This means that a, which is of the form 4n 3, cannot be expressed as a product of primes of the form 4m 1, leading to a contradiction.
Now let's consider the case where all prime divisors of a are of the form p = 2. Since a = 4n 3, it is not divisible by 2^2, so its prime factorization must be a product of 2's. However, we can easily see that no product of powers of 2 can give us a number of the form 4n 3, leading to another contradiction.
Therefore, we can conclude that a must have a prime divisor of the form p = 4m 3 for some m ∈ z, as required.
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Since a is odd, it must be of the form a = 4n + 1. Let a = 4n + 1 = p1^a1 · p2^a2 · · · pk^ak be the prime factorization of a. Suppose all prime factors of a are of the form 4m + 1. Then a ≡ 1 (mod 4), which is a contradiction. Therefore, a must have a prime factor of the form 4m + 3.
We prove the contrapositive. Suppose a has no prime divisor of the form p = 4m + 3. We show that a is not of the form a = 4n + 3.
Let a = 4n + 3. Since a is odd, it must have a prime divisor p. Note that p cannot be 2. Also, p cannot be of the form p = 4m + 3, since we assumed a has no such prime divisor. Therefore, p must be of the form p = 4m + 1.
Write a = pk, where k ∈ Z. Then 4n + 3 = pk. Since p is odd, we have 4n ≡ −3 (mod p). Squaring both sides, we get 16n^2 ≡ 9 (mod p).
Now note that 16 ≡ 1 (mod p) and so 16^(p-1) ≡ 1 (mod p) by Fermat's Little Theorem. Therefore, we have
9 = 16n^2 · 16^−2 ≡ n^2 (mod p).
This means that n^2 ≡ 9 (mod p), so p must divide (n−3)(n+3). Since p is of the form 4m + 1, neither n−3 nor n+3 is divisible by p. Therefore, p must divide both n−3 and n+3. This means that p divides their difference, which is 6. Since p is of the form 4m + 1, it cannot divide 2 or 3. Therefore, p must be 5.
But this means that a = pk is divisible by 5, which contradicts the fact that a has no prime divisor of the form 4m + 3. Therefore, we conclude that a cannot be of the form a = 4n + 3.
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find the m and b of the table
9514 1404 393
Answer:
m = 1/2
b = -7
Step-by-step explanation:
The formulas for these can be used.
m = (y2 -y1)/(x2 -x1) . . . . . . . . . formula for slope given 2 points
m = (-9 -(-11))/(-4-(-8)) = 2/4
m = 1/2
__
b = y1 -m·x1 . . . . . . . . . . . . formula for y-intercept given slope and point
b = -11 -(1/2)(-8) = -11 +4
b = -7
CAN SOMEONE HELP LIKE RN I NEED IT QUICK
Si 2X -7 = 3, luego 3X + 1 =
(1) 4
(2)5
(3)7
(4)12
(5)16
Answer:
2x-7=3
2x=3+7
{2x=10}÷2
x=5
꧁꧁꧂꧂
3x+1=
{3x=1}÷3
x=1/3
59:
Evaluate the function requested. Write your answer as a fraction in lowest terms.
B
39
15
36
Find tan A
a
5
tan A =
13
C.
5
tan A -
12
b.
13
d.
12
tan A =
tan A -
5
Answer:
Step-by-step explanation:
PLEASE HELP!!
The cost to rent a chain saw for two days and a brush chipper for two days is $272. The cost to rent a
chain saw for two days and a brush chipper for one day is $195. Write a system of equation to solve to
find the cost of renting a chain saw for one day.
the sum of two consecutive multiples of 4 is 36 . find the multiples ???
PLZZZZZZZZZ help me now
Answer:
The numbers are 16 and 20
Step-by-step explanation:
Let the multiples be: x+4 and x+8
Sum is 36, so we can make an equation:
x+4+x+8=36
2x=36-12
x=12
The numbers are 12+4=16, 12+8=20.
Hope this answer helped you! If so, please leave a rating and let me know if I was correct. (Hopefully Brainliest, but totally up to you.) Have a great day!
Write your question here (Keep it simple and clear to get the best answer) simplify (3x-4y)(5x+2y)-(x+y)
Answer:
15x^2−14xy−8y^2−x−y
Step-by-step explanation:
Combine like units.
Answer:
hope it helped
have a good day
Vincent had 97 rocks in his collection. He found 48 more at the beach. How many did he have in all ?
Answer:
145
Step-by-step explanation:
97+48
7+8=15
9+4+1=14
145
Find the measure......
Answer:
the answer would be S/ 2= 140/ 2 = 70°
You get scores of 72 and 88 on 2 science tests. You have to have an average of at least 85 to not be grounded by your parents. What score do you need to make on your next test to not be grounded?
Answer:
95
Step-by-step explanation:
72+88+95=255:3= 85
A newspaper invited readers to call a telephone number and record their comments. The next day they reported that readers overwhelmingly prefer two-way traffic flow to one-way streets. This is an example of: A. A convenience sample B. A self-selected sample C. A systematic sample D. A random sample does anyone know the answer
Answer: Self selected sample
Step-by-step explanation:
They people chose themselves
This is an example of: B. A self-selected sample , Option B is the answer.
What is a self-selected sample ?A self selected sampling technique is one in which the researcher puts out a call or publicises so that individuals and organisations can take part in the survey on their own will which saves the time for searching for the appropriate people .
It is given that
A newspaper invited readers to call a telephone number and record their comments.
This is an example of: B. A self-selected sample
Therefore Option B is the answer.
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PLEASE HELP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer: 8463160
Step-by-step explanation:
Answer:
8463160
Step-by-step explanation:
https://byjus.com/rounding-to-the-nearest-tenth-calculator/
18,000 divided by 25%
Answer:
4500
Step-by-step explanation:
18000 / 100 =180
180*25 = 4500
A leisure aircraft is owned by the Broadland Flying Group which has 10 members. The costs of keeping the aircraft comprise maintenance costs (including insurance) and the running cost which is mainly fuel. Average fuel consumption is 9 miles per gallon. If the 10 members of the Group are to share costs equally, write down a formula to enable Gordon the treasurer to calculate the annual cost for each member if the aircraft travels n miles in a year, incurs maintenance costs of £M and fuel costs £3 a gallon.
Answer:
£(M + n/3) / 10
Step-by-step explanation:
Given that :
Number of members = 10
Fuel consumption = 9 miles per gallon
Consumption for n miles = n/9 GALLONS
Cost of fuel = n/ 9 * £3 = 3n / 9 = £n / 3
Maintainace cost = £M
Annual cost per member :
£(M + n/3) / 10
Express the function F in the form f ∘ g. (Enter your answers as a comma-separated list. Use non-identity functions for f(x) and g(x).) F(x) = x + 2
Answer:
[tex]F(x) = f \circ g[/tex]
[tex]f(x) = x + 2[/tex] and [tex]g(x) = x[/tex]
Step-by-step explanation:
Given
[tex]F(x) = x + 2[/tex]
Required
Express as
[tex]F(x) = f \circ g(x)[/tex]
In functions:
[tex]f \circ\ g(x) = f(g(x))[/tex]
So, we have:
[tex]f(g(x)) = x + 2[/tex]
g(x) can be set to x and f(x) to x + 2
Hence:
[tex]F(x) = f \circ g[/tex]
When: [tex]f(x) = x + 2[/tex] and [tex]g(x) = x[/tex]
HELP ASAP! EXPONENTIAL FUNCTIONS
1. Which of the following equations would NOT contain the point (4,12)?
A. y=-1/2x+14
B. y=1/2x+8
C.y=-2x+20
D.y=3/4x+9
Answer:
the answer is b hope this helps :)
Jack made 9 identical bags using 5 yards of fabric. How much fabric did Jack use for each bag? Solve on paper and check your work on Zearn.
Answer: 5/9 yards
Step-by-step explanation:
From the question, we are informed that Jack made 9 identical bags using 5 yards of fabric.
To calculate the amount of fabric that Jack use for each bag, we have to divide 5 by 9. Thus will be:
= 5 / 9
Therefore, it means that Jack used 5/9 yards for each bag
coat is 20% off. If the regular price of the coat is $225, what is the sale price of the coat?
Answer:
$180
Step-by-step explanation:
$225 times .20 = 45 . 225-45=180
Answer:
$180
Step-by-step explanation:
Find 1% of the price.
100% = 225
1% = 225 ÷ 100 = 2.25
100 - 20 = 80% (Percentage of sale price)
80% = 2.25 x 80 = 180
Find the area of the circle 18in
1017.88in^2
You need to use pi
Answer:
1017.88inches^2
Step-by-step explanation:
Area of a circle=pi * radius^2
324*pi=1017.88
.
Simplify the radical expression.
−4x2
2x2
−2x2
4x2
Plz help me with 18 is not -0.95
Answer:
Please refer to the attachment.......
Micah invests $5,280 in an account that earns 4.2% interest, compounded monthly.
Part A
Write a function for the amount that will be in the account after t years.
Part B
What is the value of the account after 8 years?
a) The function is given as follows: [tex]A(t) = 5280\left(1 + \frac{0.042}{12}\right)^{12t}[/tex]
b) The value of the account after 8 years is given as follows: $7,384.
What is compound interest?The amount of money earned, in compound interest, after t years, is given by:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
In which:
P is the principal, which is the value of deposit/loan/....r is the interest rate, as a decimal value.n is the number of times that interest is compounded per year, annually n = 1, semi-annually n = 2, quarterly n = 4, monthly n = 12.The parameter values for this problem are given as follows:
P = 5280, r = 0.042, n = 12.
Hence the equation is given as follows:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]A(t) = 5280\left(1 + \frac{0.042}{12}\right)^{12t}[/tex]
The value of the account after 8 years is obtained as follows:
[tex]A(8) = 5280\left(1 + \frac{0.042}{12}\right)^{12 \times 8}[/tex]
A(8) = $7,384.
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I NEED HELP ASAP WILL GIVE BRAINLYEST
Answer:
-8
Step-by-step explanation:
First simplify the expression to -(2^5)^3/5. Then you get -2^3, which is simplified again to -8.
graph the linear equation y = -6x - 5
this was my answer from demos
What is the solution to the equation 3^x = 10?
Answer:
log3 (10) = x
the 3 is a tiny 3
Step-by-step explanation:
remember the saying, left right left.
so, 3 goes to the left, x goes to the right, and 10 goes to the left side of the equal sign which in turn comes out to log3(10) =x
Find the volume of the cylinder.
Either enter an exact answer in terms of it or use 3.14 for
how muck is in kilograms?
Answer:
1000grams= 1 kilogram
Step-by-step explanation:
Answer:
1000 grams = 1 Kilo
Dont be shy- drop the answer :)
Answer:
-11x = -55
x = 5
Step-by-step explanation:
Hope that helps!
A 2-year loan of $500 is made with 4% simple interest.Find the interest earned
Answer:
$40
Step-by-step explanation:
Interest = Principal x Rate x Time
I = 500(.04)(2)
Based on the information given the interest earned is $40.
Using this formula
Interest =Principal× Rate× Time
Where:
Time =2 year
Principal=$500
Rate=4% or 0.04
Let plug in the formula
Interest=500(0.04)(2)
Interest=$40
Inconclusion the interest earned is $40.
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