Show that if p is an odd prime , then 2(p-3)! congruent to -1
(mod p)
Answers
We have shown that 2(p-3)! is congruent to -1 (mod p) when p is an odd prime. 2(p-3)! ≡ -1 (mod p).
To show that 2(p-3)! is congruent to -1 (mod p) when p is an odd prime, we can apply Wilson's theorem, which states that for any prime number p, (p-1)! is congruent to -1 (mod p).
First, let's consider (p-1)!. According to Wilson's theorem, (p-1)! is congruent to -1 (mod p) for any prime p.
Now, let's look at (p-3)!:
(p-3)! = (p-1)! / [(p-2)(p-1)]
Since (p-1)! is congruent to -1 (mod p) according to Wilson's theorem, we can rewrite the equation as:
(p-3)! ≡ -1 / [(p-2)(p-1)] (mod p)
To simplify further, we can multiply both the numerator and the denominator by 2:
2(p-3)! ≡ -2 / [(p-2)(p-1)] (mod p)
Now, we need to show that -2 / [(p-2)(p-1)] is congruent to -1 (mod p). To do that, we can multiply both sides by (p-2)(p-1):
2(p-3)!(p-2)(p-1) ≡ -2 (mod p)
Since p is an odd prime, both (p-2) and (p-1) are even, which means (p-2)(p-1) is divisible by 2. Therefore, we can divide both sides by 2:
(p-3)!(p-2)(p-1) ≡ -1 (mod p)
So, we have shown that 2(p-3)! is congruent to -1 (mod p) when p is an odd prime.
In summary:
2(p-3)! ≡ -1 (mod p)
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Related Questions
4(a + 4) = 2(a + 10)
Someone solve pls.
Answers
Answer:
a=2
Step-by-step explanation:
4a+16=2a+20
2a=4
a=2
Answer:
The value of a is 2.
Step-by-step explanation:
First, you have to expand using distributive law :
\(4(a + 4) = 2(a + 10)\)
\(4a + 16 = 2a + 20\)
Next, you have to move all the common terms to one side :
\(4a - 2a = 20 - 16\)
\(2a = 4\)
Lastly, you have to solve it by dividing:
\(a = 4 \div 2\)
\(a = 2\)
Using a = 5, b = 12 and c = 13, is triangle ABC a right angle triangle? Yes or No
Answers
The triangle with sides 5, 12 and 13 is a right triangle.
How to know a right angle triangle?A triangle can be checked if it's a right triangle by using the Pythagoras theorem as follows:
Using Pythagoras theorem,
c² = a² + b²
where
c = hypotenuse sidea and b are the other legsNow let's check if the triangle with side 5, 12 and 13 is a right triangle.
Recall the longest side is the hypotenuse.
Therefore,
5² + 12² = 13²
25 + 144 = 169
Thus, the triangle with sides 5, 12 and 13 is a right triangle.
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A hypothesis can be differentiated from a theory because it is...
a. a specific prediction arising from the theory. c. it talks about how one specific variable affects d. all of the above.
Answers
A hypothesis can be differentiated from a theory because it is a specific prediction arising from the theory. The statement that is true is, a hypothesis can be differentiated from a theory because it is a specific prediction arising from the theory. A hypothesis is a suggested explanation of a phenomenon or observed data that is testable.
Hypotheses are more detailed, and are written to explain precisely what you think is going to occur in your research and the reason for that prediction. A hypothesis will be rejected if it does not fit the data, whereas a theory will be modified to fit the data.
A hypothesis is more like a forecast, and a theory is more like a law that explains how certain events operate. Thus, we can conclude that a hypothesis is a specific prediction arising from the theory.
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Rory records the percentage of battery life remaining on his phone throughout a day. The battery life decreases as Rory uses the phone, but will increase or stay at 100% while charging. The graph represents the percentage of battery life remaining after a certain number of hours.
A graph titled Phone Battery Life. The horizontal axis shows Elapsed Time (hours) numbered 2 to 20, and the horizontal axis shows Battery Life (%) numbered 10 to 120. A line begins at 100% in 0 hours, to 20% in 8 hours, to 100% from 10 to 12 hours, to 60% in 16 hours, to 100% from 17 to 20 hours.
At which times could Rory's phone have been plugged into the charger? Select three options.
6 hours
9 hours
11 hours
14 hours
19 hour
Answers
Answer:
9, 11, and 19
Step-by-step explanation:
9, 11, and 19 are the only plausible answers because there were only 2 periods where the phone was at 100% (other than when the graph started) and these are the only hours that fell within those periods..
helpp me pleasee ╹◡╹
Answers
Answer:
It should cost you 5$
Step-by-step explanation:
10$ = 4 cans of soda
If we take half of 4 which equals 2
You should also take half of of 10$ which is 5$
So, 2 Cans= 5$
Answer:
A: 5
Step-by-step explanation:
So if a 4 pack is worth $10, then if you divide 4 in half to get 2, you also have to divide the $10 by 2. Then you have your answer! Good luck in your math career!
Krystal left the bookstore traveling 18 mph. Then, 3 hours later, Kori left traveling the same direction at 27 mph. How long until Kori catches up with Krystal
Answers
Answer:
Korl will catch up to Krystal in 2 hours.
Step-by-step explanation:
18x3=54 this shows how long Krystal has driven in 3 hours.
27x2=54 this shows that in 2 hours Korl will have driven 5 mph.
Now as you can see both quotient are 54 and the question is How long until Korl catches up to Krystal so it's 2 hours
HOPE THIS HELPED:)
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
please help me and give all steps to it
Answers
Answer:
24501.42 m³
Step-by-step explanation:
You want the volume of a cylinder that is 34 m in diameter and 27 m high.
VolumeThe volume of a cylinder is given by the formula ...
V = πr²h
where r is the radius, half the diameter, and h is the height.
Use the given values in the formula and do the arithmetic. The volume is about ...
24501.42 m³
__
Additional comment
The answer shown has 7 significant figures. The value used for π has 3 significant figures, so any figures in the answer after the first 3 are incorrect. When a more accurate value of π is used, the answer rounds to 24513.85 m³.
Meagan wants to make 2 1/3 batches of muffins. The recipe calls for 1 1/2 cups of sugar per batch. How much sugar will she need?
Answers
Answer:
1 5/9 OR 14/9
Step-by-step explanation:
First we need to find out what the question is asking. How much sugar will she need if she wants to make 2 1/3 batches?
Next, we must see if we need to divide, subtract, multiply, or add. We need to divide.
2 1/3 divided by 1 1/2
Convert mixed numbers to improper fractions: 2 1/3=7/3 1 1/3=3/2
Apply the fraction rule a/b/c/d=a*d/b*c: 7*2/3*3
14/9 OR 1 5/9
Solve by using Substitution: 3x-y=13 2x +2y = -10
Answers
Ali and ben divide £35 in the ratio of 2:5 how much do they both get
Answers
Ali will receive £10, and Ben will receive £25 when they divide £35 in the ratio of 2:5.
How to find he ratio of 2:5 how much do they both getTo divide £35 in the ratio of 2:5, we first need to calculate the total parts of the ratio:
Total parts = 2 + 5 = 7
Next, we divide the total amount (£35) into 7 equal parts:
£35 / 7 = £5
Now, we can allocate the amounts to Ali and Ben based on the ratio:
Ali's share = 2 parts * £5 = £10
Ben's share = 5 parts * £5 = £25
Therefore, Ali will receive £10, and Ben will receive £25 when they divide £35 in the ratio of 2:5.
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The population of a municipality before 2 years was 60,000 and the rate of annual growth of population is 2%. If the numbers of in-migrants and out-migrants at the end of 2 years were 750 and 410 respectively and 620 people died within this time interval, find the present population of the municipality.
Answers
Answer:
62,144
Step-by-step explanation:
60,000 x (1 + 2%)^2 : growth in 2 years
+ 750: in migrants
- 410: out migrants
- 620: died
Answer:
62,144
Step-by-step explanation:
\(60,000 \times (1 + 2\%)^{2} \)
➡️ \( = 62,424\)
In-migrants:
➡️ \(750 - 410 = 340\)
now:
\(62,424 + 340 - 620\)
➡️ \( = 62,144\)
So, the present of the municipality is 62,144
there are nine different positions on a baseball team. if a team has 15 players how many different line-ups can the team make? (assume every player can play every position.) the team can make different line-ups. baseball games consist of nine innings. a team wants to change its line-up every inning. if no game goes to extra innings, and a season consists of 66 games, how many complete seasons can the team play without repeating a line-up?
Answers
There are 18 seasons and the number of different line-ups the team can make is 24310
A) We are given the team has 17 players and there are 9 different positions in the team.
This is a combination question and thus, the number of different line-ups the team can make is;
C(17, 9) = 24310
B) We are told the basketball games consist of 9 innings.
Thus,
Number of games you can fill = 24310/9 = 2701.1111
And then we are told a season consists of 147 games.
Thus;
The number of complete seasons can the team play without repeating a line-up = 2701.1111/147 ≈ 18 seasons.
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which of the following statement is assigning the value 1.667 to array, fractions, element 9?a. fractions[2]b. fractions[5]c. fractions[4]d. fractions[3]
Answers
The correct statement that assigns the value 1.667 to the array "fractions" at element 9 is not included in the options given.
It is important to note that arrays start counting from 0, so the 9th element would be at index 8. If there was a statement like "fractions[8] = 1.667", then that would be the correct answer.
However, since none of the options include the correct index, none of them are assigning the value 1.667 to the 9th element of the "fractions" array.
None of the options have the index of 9, and therefore cannot assign a value to that specific element
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Select the prime number
73, 74, 75, 76, 77
Answers
Answer:
75 is the prime number if I am correct
I NEED HELP HURRY PLS
Answers
Answer:
I hope this helps....
Step-by-step explanation:
not quite sure if it's 100%correct.....or what ur looking 4..
What mass of titanium displaces 125.0 cm3, if the density of titanium is 4.51 g/cm3?
Answers
The titanium has a mass value of 563.75 grams
How to determine the mass of the titanium?From the question, the given parameters are
Density of titanium = 4.51 g/cm^3
Volume of benzene = 125.0 cm^3
The mass of the titanium is then calculated using
Density = Mass/Volume
Make Mass the subject of the formula
So, we have
Mass = Volume x Density
Substitute the known values in the above equation
So, we have the following equation
Mass =125.0 x 4.51
Evaluate the product in the equation
Mass = 563.75
Hence, the mass of the titanium is 563.75 grams
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n homeroom, 3 of the 16 girls have red hair and 2 of the 15 boys have red hair. what is the probability of selecting a boy or a red-haired person as homeroom representative to student council
Answers
To calculate the probability of selecting a boy or a red-haired person as a homeroom representative to student council, we need to find the probability of each event and then use the formula P(A or B) = P(A) + P(B) - P(A and B).
There are 31 students in total (16 girls + 15 boys). The probability of selecting a boy is 15/31.
There are 5 red-haired students (3 girls + 2 boys). The probability of selecting a red-haired person is 5/31.
However, we need to account for the overlap between boys and red-haired students. There are 2 red-haired boys, so the probability of selecting a red-haired boy is 2/31.
Now we can use the formula: P(boy or red-haired) = P(boy) + P(red-haired) - P(boy and red-haired) = (15/31) + (5/31) - (2/31) = 18/31.
So, the probability of selecting a boy or a red-haired person as a homeroom representative to student council is 18/31.
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A grocery store sells a bag of 3 oranges for $1.23. If Latanya spent $3.28 on oranges, how many did she buy?
Answers
Answer: 8 Oranges
Step-by-step explanation:
Given information
3 Oranges = $1.23
Total cost = $3.28
Determine the unit price of an orange
Unit price = Cost ÷ Number of Oranges
Unit price = 1.23 ÷ 3
Unit price = $0.41 / orange
Determine the number of oranges bought
Number of orange × Unit price = Total cost
N × (0.41) = (3.28)
Divide 0.41 on both sides
N = 3.28 ÷ 0.41
\(\Large\boxed{Number~of~oranges=8}\)
Hope this helps!! :)
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HELP!!
If the product of two numbers is positive, which of the following expressions must be true?
1. The factors had different signs.
2. One of the factors was negative.
3. One of the factors was positive.
4. The factors had the same sign.
Answers
Answer:
4: The factors had the same sign
Step-by-step explanation:
They are both positive (i think)
how would i go about simplifying (x+h)^3?
Answers
2(x+5) = 3x+1
How to solve
Answers
Answer: x=9
Step-by-step explanation: 2(x+5+)=3x+1 or 2x+10=3x+1
Answer:
x = 9
Step-by-step explanation:
2(x + 5) = 3x + 1 Distribute the 2
2x + 10 = 3x + 1 Subtract 2x from both sides
10 = x + 1 Subtract from both sides
9 = x
Check:
2(x + 5) = 3x + 1
2(9 + 5) = 3(9)+1
2(14) = 27 + 1
28 = 28
Estimate the distance you can travel in 4 hours 55 minutes if you drive on average 51 mile per hour. Round your answer to the nearest mile
Answers
We are given the time taken to travel a certain as an estimated 4 hours 55 minutes. Also, the average speed for the journey is given as 51 miles per hour. To calculate the distance, the formula given is;
\(Dis\tan ce=Speed\times time\)Note that the time is given as 4 hours plus a fraction of an hour, that is 55 minutes. To express this as a decimal we take the value of 55 minutes as a fraction of one hour, that is;
\(\begin{gathered} 55\min s=\frac{55}{60}hr \\ 55\min s=0.9167hr \end{gathered}\)The time taken therefore can be expressed as;
\(4.9167\text{hrs}\)The distance covered therefore would be;
\(\begin{gathered} \text{Distance}=\text{Speed x Time} \\ D=51\times4.9167 \end{gathered}\)We simplify and we now have;
\(D=250.7517\)Rounded to the nearest mile, that is, the nearest whole number;
ANSWER:
\(D=251\text{ miles}\)In circle F with m∠EHG = 35, find the m∠EFG
Answers
Answer: m∠EFG = 70
Step-by-step explanation:
m∠EFG = 2x35
m∠EFG = 70
In conclusion the answer is 70
find the z value for 0.99 if e O¨zlem likes jogging 3 days of a week. She prefers to jog 3 miles. For her 95 times, the mean wasx¼ 24 minutes and the standard deviation was S¼2.30 minutes. Let μ be the mean jogging time for the entire distribution of O¨zlem’s 3 miles running times over the past several years. How can we find a 0.99 confidence interval for μ?.
likes jogging 3 days of a week. She prefers to jog 3 miles. For her 95 times, the mean wasx¼ 24 minutes and the standard deviation was S¼2.30 minutes. Let μ be the mean jogging time for the entire distribution of O¨zlem’s 3 miles running times over the past several years. How can we find a 0.99 confidence interval for μ
a) What is the table value of Z for 0.99? (Z0.99)? (b) What can we use for σ ? (sample size is large) (c) What is the value of? Zcσffiffin p (d) Determine the confidence interval level for μ.
Answers
a. The table value of z 0.99 is given as 2.58.
b. we use σ = 2.30 minutes.
c. The value of Z is given as 0.609.
d. The confidence interval is (23.391, 24.609) minutes.
How to solve for the z critical valuea) To find the z-score for a 0.99 confidence interval, we need to find the z-score that leaves 0.5% (since it's two-tailed, we split the 1% or 0.01 of the data that's not within the confidence interval into two) in each tail. The z-score for 0.995 (0.5% in the upper tail) is approximately 2.58 in standard normal distribution tables. So, Z_0.99 is 2.58.
b) Therefore, we use σ = 2.30 minutes.
c) The standard error of the mean (SE) is given by σ/√n, where n is the sample size.
Here, σ = 2.30 minutes and n = 95.
Therefore, SE = σ/√n
= 2.30/√95
= 0.236 minutes.
Multiply this by the z-score to find Z * σ/√n.
Therefore, Z * σ/√n = 2.58 * 0.236 ≈ 0.609.
d) The 0.99 confidence interval for μ
Here, x is the sample mean which is 24 minutes.
So the confidence interval is approximately
= (24 - 0.609, 24 + 0.609)
= (23.391, 24.609) minutes.
This is the range of values we are 99% confident that the true population mean (μ) falls in.
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For a certain horse race, the odds in favor of a certain horse finishing in second place are given as 29 to 71.
Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive. The probability is
.(Round to two decimal places as needed.)
Answers
The probability of the horse finishing in second place is approximately 0.29.
To convert the odds given as "29 to 71" into a probability value, we can divide the number of favorable outcomes (29) by the total number of possible outcomes (29 + 71 = 100).
The probability is calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes.
In this case, the probability is 29/100, which is equal to 0.29.
To convert odds to probability, we divide the number of favorable outcomes (29) by the total number of outcomes (29 + 71 = 100). The probability of the horse finishing in second place is 29/100, which simplifies to 0.29 or 29%.
This means there is a 29% chance or likelihood of the horse finishing in second place based on the given odds.
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Rahul wants to cover his desk with chocolate bars. His desk is 1 m long and 60 cm wide. Each chocolate bar is 10 cm long and 5 cm wide. He buys 35 chocolate bars. How many more chocolate bars does he to cover his entire desk
Answers
Raul's desk has the dimension of a rectangle with an area of 6000 cm² and he will need 85 more chocolate bars to cover his entire desk.
How to calculate for the bars if chocolate.Raul's desk is a rectangle with length 1 m and width 60 cm, also the chocolate bars have length of 10 cm and width of 5 cm.
To know how many more chocolate bars Raul needs to cover his entire desk, having 35 already, we divide the area of Raul's desk by area of a chocolate bar, and thus subtract 35 from the result.
1 m = 100 cm
area of Raul's desk = 100 cm × 60 cm
area of Raul's desk = 6000 cm²
area of a chocolate bar = 10 cm × 5 cm
area of a chocolate bar = 50 cm²
number of chocolate that will cover the entire desk = 6000 cm²/50 cm²
number of chocolate that will cover the entire desk = 120
Therefore, since Raul bought only 35 chocolate bars, he will need 85 more chocolate bars to cover the entire desk having an area of 6000 cm².
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A camera shop stocks six different types of batteries, one of which is type A7b. Suppose that the camera shop has only twelve A7b batteries but at least twenty of each of the other types. Now, choose the correct answer for the following question - How many ways can a total inventory of twenty batteries be distributed among the six different types? C(24, 20) - C(12,7) C(25, 20) - C(12,7) C(20, 15) - C(13,7) C(25, 20) - C(13,7)
Answers
D: C(25, 20) - C(13,7) represents the number of ways a total inventory of twenty batteries be distributed among the six different types.
To find the total number of ways to distribute the twenty batteries among the six different types, we need to use the stars and bars method. Using this method, the number of ways to distribute 20 batteries among 6 different types is C(20 + 6 - 1, 6 - 1) = C(25, 5).
However, since we have only 12 A7b batteries, we need to subtract the cases where we distribute more than 12 A7b batteries. To do this, we calculate the number of ways to distribute the remaining 8 batteries among the 5 types other than A7b, which is C(8+5-1, 5-1) = C(12,4). Therefore, the total number of ways to distribute 20 batteries among the six different types while keeping at least 12 A7b batteries is C(25, 5) - C(12, 4) = C(25, 20) - C(13, 7).
The correct answer is option D.
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A piece of climbing equipment at a playground is 6 feet high and extends 4 feet horizontally. A piece of climbing
equipment at a gym is 10 feet high and extends 6 feet horizontally. Which statement best compares the slopes of the
two pieces of equipment?
Answers
Answer:
First Option i.e 5/3 > 3/2 , the slope of Climbing equipment at Gym is greater is Correct
Step-by-step explanation:
This question deals with Slope of a line.
Slope tells how vertical a line is.
The more the slope is, the more the line is vertical. When slope is zero, the line is horizontal.
To find the slope, we take the ratio of how much the line's height increases as we go forward or backward on the horizontal axis.
For first climbing equipment rise = 6 feet, run(horizontal) = 4 feet
Thus, slope = 6/4 = 3/2 ≈ 1.5
For second climbing equipment rise = 10 feet, run(horizontal) = 6 feet
Thus, slope= 10/6 = 5/3 ≈ 1.66
Thus,
Slope of first climbing equipment = 3/2 < slope of second climbing equipment = 5/3
Thus, First Option i.e 5/3 > 3/2 , the slope of Climbing equipment at Gym is greater is Correct
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1. how much do national football league (nfl) players weigh, on average? in a random sample of 50 nfl players, the average weight is 244.4 pounds.
Answers
The average weight of a random sample of 50 NFL players is 244.4 pounds.
What is an equation?An equation shows the relationship between two or more numbers and variables.
For a samples size (n) in normally distributed population, the sample mean is normally distributed, with mean μX=μ and standard deviation σX=σ/√n.
Hence for a sample of 50 NFL players with an average weight of 244.4 pounds:
Mean = mean = 244.4 pounds
The average weight is 244.4 pounds.
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Which expression represents the number -2i(5- i) + (17- 8i) rewritten in a + bi
form?
O 15-18i
O 15-2i
O 19 - 18i
O 11 + 8i
Answers
Answer:
(a) 15 -18i
Step-by-step explanation:
You want the simplified form of the expression -2i(5- i) + (17- 8i).
Complex numbersFor many purposes, the value i in a complex number can be treated in the same way a variable would be treated. When simplifying an expression involving i, any instances of i² can be replaced with the real value -1.
-2i(5- i) + (17- 8i) = -10i +2i² +17 -8i
= -2 +17 +(-10 -8)i
= 15 -18i
__
Additional comment
Your scientific or graphing calculator can probably help you evaluate such expressions.
