Demonstrate that the equation x^10+y^10-z^10 = 5 has no integer solutions.
Answers
The equation\( x^10+y^10-z^10 = 5\) has no integer solutions, we can use a proof by contradiction. Assume that there exists a solution to this equation in integers x, y, and z.
Without loss of generality, we can assume that x, y, and z are relatively prime (otherwise, we can divide all three by their greatest common divisor).
Notice that x^10 and y^10 are always nonnegative, so their sum is at least 0. Therefore, we have
\( z^10 <= x^10 + y^10 + 5. \)
Taking the equation mod 3, we see that x^10, y^10, and z^10 are all congruent to either 0 or 1 mod 3. If any of them is congruent to 0, then we can divide all three by 3, contradicting the assumption that they are relatively prime. Therefore, all three are congruent to 1 mod 3.
But this means that\( x^10 + y^10 + 5\) is congruent to 2 mod 3, while z^10 is congruent to 1 mod 3. This is a contradiction, so our assumption that there exists a solution to the equation in integers x, y, and z must be false. Therefore, the equation x^10+y^10-z^10 = 5 has no integer solutions.
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Related Questions
5 plus 5 equals what?
Answers
Answer: 10
Step-by-step explanation: 1 1 1 1 1+ 1 1 1 1 1= 1 1 1 1 1 1 1 1 1 1 or 10
A=bh slove for b
jnbirnbiunrtortn
Answers
Answer:
b = A/h
Step-by-step explanation:
A = bh
Divide both sides by h:
A/h = bh/h
A/h = b
The circumference of the inner circle is 270.04 miles. What is the area of the shaded region?
Answers
area of a circle = \(\pi r^{2}\)
circumference =\(2\pi r\)
Okay so the center of the circle is the same for both of the circles lets take that value as m so the radius of the inner circle is 54 - m
we have the circumference of the inner circle which is 270.04 we can find the value of m using that
\(270.04=2\pi (54-m)\\270.04= 339.2-6.28m \\6.28m = 339.2-270.04\\m= 11.02\)
the radius of the inner circle is therefore 42.98
and the full circle has a radius off 65.02
area of shaded region = area of full circle - area of inner circle
area of shaded region = \(\pi (65.02)^{2} - \pi (42.98)^{2}\)
area of shaded region = \(13281.39 - 5803.4\)
area of shaded region = 7478 rounded off to the nearest whole number
andrea and lauren are 20 kilometers apart. they bike toward one another with andrea traveling three times as fast as lauren, and the distance between them decreasing at a rate of 1 kilometer per minute. after 5 minutes, andrea stops biking because of a flat tire and waits for lauren. after how many minutes from the time they started to bike does lauren reach andrea? (a) 20 (b) 30 (c) 55 (d) 65 (e) 80
Answers
The correct answer is option (C) 55 minutes.
Let's break down the information given in the problem:
Distance between Andrea and Lauren at the start: 20 kilometersRate at which the distance between them is decreasing: 1 kilometer per minuteAfter 5 minutes, Andrea stops biking and waits for LaurenTo find out how long it takes for Lauren to reach Andrea, we need to determine the time it takes for Andrea to cover half the distance between them. Once Andrea reaches the halfway point, Lauren will also have traveled the same distance.
Let's denote the time it takes for Andrea to reach the halfway point as "t" minutes. Since the distance decreases at a rate of 1 kilometer per minute, after "t" minutes, the distance between Andrea and Lauren will be reduced by "t" kilometers.
Now, let's calculate the distance traveled by Andrea in "t" minutes:
Distance = Rate × Time
Since Andrea is traveling three times as fast as Lauren, her rate is 3 times the rate of Lauren.
Distance traveled by Andrea in "t" minutes = (3 × 1) × t = 3t kilometers
The total distance between Andrea and Lauren at that time will be:
Distance between Andrea and Lauren = 20 - t kilometers
Since they meet at the halfway point, the distance traveled by Andrea (3t) will be equal to half the total distance (20 - t)/2:
3t = (20 - t)/2
To solve this equation, we can multiply both sides by 2:
6t = 20 - t
Now, solve for "t":
6t + t = 20
7t = 20
t = 20/7
Therefore, it will take Andrea 20/7 minutes to reach the halfway point.
Since Lauren continues biking for 5 more minutes after Andrea stops, the total time it takes for Lauren to reach Andrea is:
Total time = t + 5 = 20/7 + 5
To calculate this, we can convert 5 minutes to a fraction with a denominator of 7:
5 minutes = 35/7 minutes
Total time = 20/7 + 35/7 = (20 + 35)/7 = 55/7
So, Lauren will reach Andrea after 55/7 minutes.
To find the answer option that matches this time, we can calculate 55/7 and compare it to the answer choices:
55/7 ≈ 7.86
Among the given answer choices, the closest option to 7.86 is (C) 55. Therefore, the answer is (C) 55 minutes.
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Five hats and three shirts cost $176. Three hats and five shirts cost $208. What is the cost of one hat?
The cost of one hat is __ dollars.
Answers
Let hats be x
Let shirts be y
We can set up a system of equations to figure out the cost of one hat.
5x+3y = 176 and 3x+5y = 208
Since we are finding the cost of the hats, we will need to isolate x from the equation. We can first get rid of y. One way we can do that is by multiplying the first equation with 5 and the second one with -3, effectively making y add up to 0 when we solve for x.
5(5x+3y = 176) --> 25x+15y = 880
-3(3x+5y = 208) --> -9x-15y = -624
We can now add the 2 equations together and solve for x.
25x+15y = 880
-9x-15y = -624
+>>>>>>>>>>>>>>>
16x = 256
x = 16
So the the cost of one hat is 16 dollars.
Answer:
$16
Step-by-step explanation:
Define the variables:
Let x = cost of one hatLet y = cost of one shirtCreate two equations using the defined variables and the given information.
Equation 1
Five hats and three shirts cost $176.
⇒ 5x + 3y = 176
Equation 2
Three hats and five shirts cost $208.
⇒ 3x + 5y = 208
Rewrite Equation 2 to make y the subject:
\(\implies \sf 3x+5y-3x=208-3x\)
\(\implies \sf 5y=208-3x\)
\(\implies \sf \dfrac{5y}{5}=\dfrac{208-3x}{5}\)
\(\implies \sf y=\dfrac{208-3x}{5}\)
Substitute the found expression for y into Equation 1 and solve for x:
\(\implies \sf 5x+3\left(\dfrac{208-3x}{5} \right)=176\)
\(\implies \sf 5x+\dfrac{624-9x}{5}=176\)
\(\implies \sf 5x+124.8-1.8x=176\)
\(\implies \sf 3.2x+124.8=176\)
\(\implies \sf 3.2x+124.8-124.8=176-124.8\)
\(\implies \sf 3.2x=51.2\)
\(\implies \sf \dfrac{3.2x}{3.2}=\dfrac{51.2}{3.2}\)
\(\implies \sf x=16\)
Therefore, the cost of one hat is 16 dollars.
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Please help me guys, I need some answers
Answers
A one is 12
B is 25
just multiply by 6
Step-by-step explanation:
Answer:
a) 24cm²
b)150cm²
c)600m²
d)864mm²
e)294m²
f)486cm²
g)384mm²
h)1014cm²
Step-by-step explanation:
Find the area of the face, it's a cube so just multiply it by the length you have
2x2=4
It has 6 faces so multiply this by 6
6x4=24
what is the answer it weird and hard
Answers
Answer:
x=12
Step-by-step explanation:
7.5/5=1.5
1.5x8=12
please help me. I think I have it figured out but I just wanted to double check.
Answers
Let's consider triangle ABC
Length AB can be obtained using Pythagoras
\(\begin{gathered} AB^2=x^2+y^2 \\ AB\text{ = }\sqrt[]{x^2+y^2} \\ \end{gathered}\)Similarly, we can consider triangle ACD, so that length AD will be obtained through Pythagoras
\(\begin{gathered} AD^2=x^2+z^2 \\ AD\text{ = }\sqrt[]{x^2+z^2} \end{gathered}\)Considering triangle ABD, with BD being the hypotenuse
\(\begin{gathered} BD^2=AD^2+AB^2 \\ (y+z)^{2\text{ }}=(x^2+z^2)+(x^2+y^2\text{)} \end{gathered}\)Expanding the parentheses
\(\begin{gathered} y^2+2yz+z^2=x^2+z^2+x^2+y^2 \\ \\ y^2-y^2+z^2-z^2+2yz=2x^2 \\ 2yz=2x^2 \\ \end{gathered}\)Divide both sides by 2
\(\begin{gathered} \frac{2yz}{2}=\text{ }\frac{2x^2}{2} \\ yz=x^2 \\ \\ x^2\text{ =yz} \\ \\ x\text{ = }\sqrt[]{yz} \end{gathered}\)Option A is correct
Answer:
Let's consider triangle ABCLength AB can be obtained using PythagorasSimilarly, we can consider triangle ACD, so that length AD will be obtained through PythagorasConsidering triangle ABD, with BD being the hypotenuseExpanding the parentheses Divide both sides by 2Option A is correct
Step-by-step explanation:
Simplify
1) 7a+ 1 + 2a + 3
Answers
. Find the slope of the line passing through these two
points: (4, 2) and (-1,6)
Answers
Answer:
m = -4/5 or m = -0.8
Step-by-step explanation:
Hope it helps:)
Which expression below is a binomial?
(1) x²-9
(3) 4x²
(2) 7x³y²
(4) 3x+2y-8z
Answers
Answer:
x²-9
Step-by-step explanation:
x²-9 has two variable .
So it is binomial.
4. What are the Z-scores for the following Confidence Interval levels? Remember, you MUST account for both tails of the curve, positive and negative, when identifying each. That means you will need to do a little math to obtain the correct z-value. 3 Points 68%= 85% = 99% =
Answers
In order to calculate the z-scores for the given Confidence Interval (CI) levels, we need to use the Z-table. It is also known as the standard normal distribution table. Here are the z-scores for the given Confidence Interval levels:1. 68% CI: The confidence interval corresponds to 1 standard deviation on each side of the mean.
Thus, the z-score for the 68% \(CI is ±1.00.2. 85% CI\): The confidence interval corresponds to 1.44 standard deviations on each side of the mean.
We can calculate the z-score using the following formula:\(z = invNorm((1 + 0.85)/2)z = invNorm(0.925)z ≈ ±1.44\)Note that invNorm is the inverse normal cumulative distribution function (CDF) which tells us the z-score given a certain area under the curve.3. 99% CI: The confidence interval corresponds to 2.58 standard deviations on each side of the mean. We can calculate the z-score using the following formula:\(z = invNorm((1 + 0.99)/2)z = invNorm(0.995)z ≈ ±2.58\)
Note that in general, to calculate the z-score for a CI level of (100 - α)% where α is the level of significance, we can use the following formula:\(z = invNorm((1 + α/100)/2)\) Hope this helps!
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each recipe of punch calls for 3 cups of lemonade per batch. corrina plans to make 20 batches of punch. how many cups of lemonade are in 20 batches of punch?
Answers
The recipe for punch calls for 3 cups of lemonade per batch. Corrina wants to make 20 batches of punch. there are 60 cups of lemonade in 20 batches of punch.
A batch requires 3 cups of lemonade. The number of cups in twenty batches of punch can be calculated by multiplying the number of cups required per batch by the number of batches made.
Hence, 20 × 3 = 60 cups of lemonade required in 20 batches of punch is 60 cups.
Therefore, there are 60 cups of lemonade in 20 batches of punch.
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A club consists of 5 girls (Kirsten, Sarah, Suzie, Monica, and Katie) and 3 boys (Kevin, Steve and Samuel). Find P(Girl | K-name)
Answer:
2/3
Answers
The total probability of a girl having a "K-name" is 3/8. The total probability of there being a girl in the club is 5/8.
Therefore, the conditional probability of having a girl with a "K-name" given that there is a girl in the club can be calculated as follows:
P (Girl | K-name) = P (K-name | Girl) × P (Girl) / P (K-name) = 2/5 * 5/8 / 3/8 = 2/3
Therefore, the probability of a girl with a K-name given that there is a girl in the club is 2/3.
In a club consisting of 5 girls (Kirsten, Sarah, Suzie, Monica, and Katie) and 3 boys (Kevin, Steve, and Samuel), we are asked to find P(Girl | K-name).
We can use Bayes' theorem to calculate the conditional probability of this event. This theorem states that the probability of an event given another event can be calculated as the product of the probability of the second event Therefore, the probability of a girl having a "K-name" given that there is a girl in the club is 2/3.
The conditional probability of there being a girl in the club given that a girl has a K-name is 2/3.
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Find the square root.
±√1.69 = ____ and ____
Answers
Answer:
±√1.69 = +1.3 and -1.3Step-by-step explanation:
√169 = 13
square root and two digits after dot means the result need to has one digit after dot (number of digits after dot divided by 2 because of square root)
so √1.69 = 1.3
A. 4x+2=6-x B. 5x+2=6 How can we get equation B to equation A?
Answers
Answer:
All you have to do on A. where there's a -x, you do the opposite to both sides of the equation or equal sign. Then you get A. from 4x+2=6-x to 5x+2=6.
Step-by-step explanation:
4x + 2 = 6 - x
+x +x
5x + 2 = 6
Hope this helps!
Answer:
Subtract 1x from both sides
B) Add/subtract the same quantity to/from both sides
Step-by-step explanation:
I glanced over both equations and noticed only one thing was really different from them bot. There was an extra x in equation A. Then I noticed if you add the 1x t both sides of equation A than it would be the same as equation B. That is when I realized that equation B needed 1x subtracted from both sides.
A) Add/subtract a quantity to/from only one side
B) Add/subtract the same quantity to/from both sides
C) Multiply/divide only one side by a non-zero constant
D) Multiply/divide both sides by the same non-zero constant
Solve |10x + 50| = 0.
Answers
Answer:
x = -5
Step-by-step explanation:
Now we have to,
→ find the required value of x.
The equation is,
→ |10x + 50| = 0
Then the value of x will be,
→ |10x + 50| = 0
→ 10x + 50 = 0
→ 10x = 0 - 50
→ 10x = -50
→ x = -50/10
→ [ x = -5 ]
Hence, the value of x is -5.
Need help plz dont be wrong
Answers
Answer:
66
Step-by-step explanation:
What is the value of x for the parallelogram shown?
A. 30
B. 75
C. 60
D. 15
Answers
Answer:
A.30
Step-by-step explanation:
^ABC = 180⁰-60⁰=120⁰
=> 4x⁰=120⁰
=> x = 30⁰
what is the ratio of their heights if bill is 5'4" tall and mary is 5'8" tall
Answers
5’8 = (5x12) + 8 = 68 inches
16:17
What is the first step in derivative classification?
Answers
Determining the level of classification based on current classification recommendations is the first step towards derivatively classifying the new document.
The main resources for determining a derived classification are Security Classification Guides (SCG).
What is derivative classification?It is possible to classify information in a new way by integrating, paraphrasing, restating, or creating it, and then classifying the new information in accordance with the original document's classification markings.
Some key features of derivative classification is-
Derivative categorisation does not include the duplication or copying of already-classified information. It is not necessary for those who use classification markings taken from sources or in accordance with a classification guide to have the original classification authority.Derivative classification has significant impacts just on Department of Defence as industry, just like original classification. Information security is improved by classifying it. It restricts access only to those people who are eligible and have a justifiable need to know the information.To know more about the derivative classification, here
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Andy runs 873.6 meters in 124.16 seconds. If Andy runs at a constant speed, how far does he run in one second?
Answers
Answer:
about 7.04 meters
Step-by-step explanation:
873.6m/124.16sec=7.04 meters per second
Answer:
If you round, he runs 7 meters in a second.
Step-by-step explanation:
pls help answer thanks
Answers
Answer:
upper right corner
Step-by-step explanation:
just let x=0
then you know y>1
so those in left is wrong
and it's> not >=
y isn't equal to 1 so bottom right corner is wrong
A simply supported beam 10 m long carries a uniformly distributed load of 24 kN/m over its entire span. E = 200 GPa, and I = 240 x 106 mm4. Compute the deflection at a point 4 m from the left support. Select one: a. 44 mm b. 75 mm c. 62 mm d. 58 mm
Answers
The deflection at a point 4 m from the left support of the simply supported beam is 44 mm.
To compute the deflection at a point 4 m from the left support of a simply supported beam, we can use the formula for deflection due to a uniformly distributed load.
First, let's calculate the value of the load acting on the beam. The uniformly distributed load of 24 kN/m is applied over the entire span of 10 m, so the total load can be found by multiplying the load per meter by the length of the beam:
Total load = 24 kN/m * 10 m = 240 kN
Next, we need to calculate the bending moment at the point 4 m from the left support. The bending moment can be determined using the formula:
Bending moment = (load per unit length * length^2) / 2
Bending moment = (24 kN/m * (4 m)^2) / 2 = 192 kNm
Now, we can calculate the deflection at the point using the formula for deflection due to bending:
Deflection = (5 * load * distance^4) / (384 * E * I)
where E is the modulus of elasticity and I is the moment of inertia of the beam.
Plugging in the values, we get:
Deflection = (5 * 240 kN * (4 m)^4) / (384 * 200 GPa * 240 * 10^6 mm^4)
Simplifying the units, we have:
Deflection = (5 * 240 * 10^3 N * (4 * 10^3 mm)^4) / (384 * 200 * 10^9 N/mm^2 * 240 * 10^6 mm^4)
Deflection = (5 * 240 * 10^3 * 4^4) / (384 * 200 * 240 * 10^9)
Deflection = 44 mm
Therefore, the deflection at a point 4 m from the left support of the simply supported beam is 44 mm.
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How does an underestimate relate to rounding factors before estimating? Explain in complete sentences. HELPPPPPP PLEASEEEEEEEEEE
Answers
Answer:
If factors are only rounded up, then the estimate is an overestimate. If factors are only rounded down, then the estimate is an underestimate. If factors are rounded up and down, it is more difficult to determine whether the estimate is an overestimate or an underestimate.
Answer:
the over
Step-by-step explanation:
HELP PLEASEE!!!!!!!!!! ASAP!!!!!!!!!!!!!!!!
Answers
Answer:
the answer is C
Step-by-step explanation:
i know because I did this
Answer:
(2,3)
Step-by-step explanation:
if the square rotated 90 degrees clockwise, it would go to the otherside of the y-axis and B' would end up on 2,3
Answer this asap 3x²+2(x+2)-3x(2x+1)
Answers
Answer: -1 (x-1) (3x+4)
Step-by-step explanation: You have to dristibute, combine Like terms, find the common factors, use the sum- product pattern, common factor from 2 pairs, re-write in correct form then you got your answer
Is the following conditional true?
If a number is rational, then it is a quotient of two integers.
Answers
Answer:
False
Step-by-step explanation:
The statement itself is wrong in the textbook is given that if a number is a rational then it si the additive inverse of the stsrement
Jordan has a garden that is enclosed by a rectangular fence. The perimeter of the garden is 84 feet. The length of one side of fencing is 8 yards. What is the area of Jordan's garden?
Answers
Answer: 432 ft^2
Step-by-step explanation:
See attached picture
Find the value of x that makes 25x^2 + 70x + c a perfect square trinomial
Answers
The value of x that makes 25x^2 + 70x + c a perfect square trinomial is c = 1225.
To make the quadratic expression 25x^2 + 70x + c a perfect square trinomial, we need to determine the value of c.
A perfect square trinomial can be written in the form (ax + b)^2, where a is the coefficient of the x^2 term and b is half the coefficient of the x term.
In this case, a = 25, so b = (1/2)(70) = 35.
Expanding (ax + b)^2, we have:
(25x + 35)^2 = 25x^2 + 2(25)(35)x + 35^2
= 25x^2 + 70x + 1225.
Comparing this with the given quadratic expression 25x^2 + 70x + c, we can see that c = 1225.
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