Using fermat's little theorem, find the least positive residue of $2^{1000000}$ modulo 17.

Answers

Answer 1

Fermat's little theorem states that
[tex]a^p[/tex]≡a mod p

If we divide both sides by a, then
[tex]a^{p-1}[/tex]≡1 mod p
=>
[tex]a^{17-1}[/tex]≡1 mod 17
[tex]a^{16}[/tex]≡1 mod 17

Rewrite
[tex]a^{1000000}[/tex] mod 17 as
[tex]=(a^{16})^{62500}[/tex] mod 17
and apply Fermat's little theorem
[tex]=(1)^{62500}[/tex] mod 17
=>
[tex]=(1)[/tex] mod 17

So we conclude that
[tex]a^{1000000}[/tex]≡1 mod 17

Answer 2

By applying Fermat's Little Theorem, we can find the least positive residue of [tex]2^{1000000[/tex] modulo 17. Using the repeated squaring method, we calculate that [tex]2^{16[/tex] is congruent to 1 modulo 17. Therefore, [tex]2^{1000000[/tex] is also congruent to 1 modulo 17.

Fermat's Little Theorem states that if p is a prime number and a is any positive integer that is not divisible by p, then a raised to the power of p-1 is congruent to 1 modulo p. In this case, we need to find the least positive residue of 2 raised to the power of 1000000 modulo 17.

First, we need to find the value of 2 raised to the power of 16 modulo 17, since 17 is a prime number. Using the repeated squaring method, we can calculate:

[tex]2^2[/tex] = 4 (mod 17)

[tex]2^4[/tex] = [tex](2^2)^2[/tex] = [tex]4^2[/tex] = 16 (mod 17)

[tex]2^8[/tex] = [tex](2^4)^2[/tex] = [tex]16^2[/tex] = 1 (mod 17)

[tex]2^{16[/tex] = [tex](2^8)^2[/tex] = [tex]1^2[/tex] = 1 (mod 17)

Now, we can find the value of [tex]2^{1000000[/tex] modulo 17 using the fact that [tex]2^{16[/tex] is congruent to 1 modulo 17:

[tex]2^{1000000[/tex] = [tex](2^{16})^{62500[/tex] x [tex]2^0[/tex] = [tex]1^{62500[/tex] x 1 = 1 (mod 17)

Therefore, the least positive residue of [tex]2^{1000000[/tex] modulo 17 is 1.

Related Questions

Megan buys 3 bracelets and 3 necklaces. Each bracelet costs 5. Megan pays on 40 and gets 4 change. what is the cost of one necklace?

Answers

The necklaces cost $3 each

In a factory a manager tests 250 products and find defcts in 7 of them how many defcts are likely going to be in 10000 unit order

Answers

So first you take 7/250*x/10000 and you cross multiply. Then you get
70000=250x
and the answer is
x=280

I hope this helps!!!!!!!

kevin had 13 coins in his pocket. all of his coins are either dimes or quarters. when he emptied his pocket, kevin found that he had 2.05. how many dimes and quarters does he have?

Answers

5 quarters and 8 dimes = $2.05

Answer:

Kevin has 5 quarters and 8 dimes.

Step-by-step explanation:

First of all, a quarter is worth 0.25 dollars, while a dime is worth 0.10 dollars. As Kevin has 13 coins in his pocket which in sum make $2.05, and all of them are either dimes or quarters, in order to determine how many dimes and quarters does he have, we have to take into account that he's got an odd number of quarters, since the final amount of money sums $2.05, that is, it ends in 5.

Therefore, if he had 1 quarter and 12 dimes (0.25 + 12x0.10 = 1.45), he would have $1.45.

If he had 3 quarters and 10 dimes (0.25x3 + 10x0.10 = 1.75), he would have $1.75.

If he had 5 quarters ahd 8 dimes (0.25x5 + 8x0.10 = 2.05), he would have $2.05, so this is the correct option.

Find the general solution to y′′+7y′=0y′′+7y′=0. give your answer as y=..

Answers

Try this option:
1. characteristic equation is:
a²+7a=0;
[tex] \left[\begin{array}{ccc}a=0\\ a=-7\end{array}\right[/tex]
2. y=C₁+C₂e⁻⁷ˣ.

Calculate the resistance of a piece of aluminum wire with a diameter of 100 mils and a length of two miles, at 68°F. Hint: Be sure to first convert mils to cmils and use the K value for aluminum found in the reference. (Round the FINAL answer to two decimal places.)

Answers

Answer: [tex]R = 16.83 \text{ ohms}[/tex]

Explanation:

The resistance of an aluminum wire is given by

[tex]R = \frac{\rho L}{A} [/tex]

where:

[tex]R = \text{ resistance of the wire} \\ \rho = \text{resistivity of the aluminum} = 2.65 \times 10^{-8} \text{ ohm-meters} \\ L = \text{length of the wire} = 2 \text{ miles} \\ A = \text{cross-sectional area of the wire} [/tex]

Since the resistivity is in ohm-meters, we need to convert the length to meters and the area to square meters. But we need to convert mils to cmils first to get the cross sectional area of the wire.

To convert mils to cmils, we get the square of the diameter of the circle in mils because 1 cmil (or circular mil) is the area of the circle whose diameter is 1 mil. So, since the diameter of the wire is 100 mils, the cross-sectional area of the wire in cmils is given by:

[tex]\text{Area in cmils = }d^2 = 100^2 = 10,000 = 10^4 \text{ cmils}[/tex]

Note that we convert the area in terms of scientific notation because the resistivity is expressed in scientific notation and it is easier to multiply very large and very small numbers if they are expressed in scientific notation.

Now, since [tex]1 \text{ cmil} = 5.067 \times 10^{-10} \text{ square meters}[/tex], the cross sectional area of the wire is equal to

[tex]10^4 \text{ cmils} = (10^4 \text{ cmils} )\left ( \frac{5.067 \times 10^{-10} \text{ square meters}}{1\text{ cmil}} \right ) [/tex]
[tex]10^4 \text{ cmils} = 5.067 \times 10^{-6} \text{ square meters}[/tex]

Since 1 mile = 1,609.344 meters, 2 miles = 2 × 1,609.344 meters = 3,218.688 meters = [tex]3.218688 \times 10^3 \text{ meters}[/tex]. .

Hence the resistance is given by

[tex]R = \frac{\rho L}{A} \\ R = \frac{(2.65 \times 10^{-8} \text{ ohm-meters}) (3.218688 \times 10^3 \text{ meters})}{5.067 \times 10^{-6} \text{ square meters}} \\ \boxed{R = 16.83 \text{ ohms}}[/tex]

Final answer:

This answer calculates the resistance of an aluminum wire of specific dimensions. It begins by determining the cross-sectional area of the wire, then uses a formula incorporating resistivity, length, and area to calculate resistance. The final resistance of the wire is approximately 0.037 Ohms.

Explanation:

To calculate the resistance of a piece of aluminum wire, we first need to figure out the cross-sectional area of the wire. As given, the diameter of the wire is 100 mils. To convert mils to cmils (circular mils), we square the diameter, thus 100^2 = 10,000 cmils is the cross-sectional area.

Next, we use the formula for resistance: R = ρL/A, where R is the resistance, ρ (Rho) is the resistivity of the material (aluminum, in this case), L is the length in feet, and A is the cross-sectional area in circular mils (cmils).

We need to convert the length from miles to feet (1 mile = 5,280 feet), so 2 miles = 10,560 feet. The resistivity of aluminum at 68°F is approximately 10.75 nanoOhms*meter, to convert this to Ohms*foot we multiply by 3.281, so ρ becomes approximately 35.28 nΩ/ft.

The resistance R can then be calculated as follows: R = ρL/A = (35.28 x 10^-9) x 10,560 / 10,000 = 0.037 Ohms, rounded to two decimal places.

Learn more about Wire Resistance here:

https://brainly.com/question/30799966

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Please help me questions 9 and 10

Answers

Problem 9's answer is choice C) rotation then reflection
Problem 10's answer is choice B) 65 degrees

---------------------------------------------------------

Explanations:

For problem 9, we rotate 90 degrees clockwise to go from figure 1 to figure 2. Then we reflect over the x axis to go from figure 2 to figure 3.

------------

For problem 10, notice how
angle AEB = angle GEC
5x = 3x+10
5x-3x = 10
2x = 10
x = 5

If x = 5, then
angle GEC = 3x+10
angle GEC = 3*5+10
angle GEC = 15+10
angle GEC = 25

So that means
angle FEG = 90 - (angle GEC)
angle FEG = 90 - 25
angle FEG = 65 degrees

What is the area of this trapezoid? Enter your answer in the box. units2 A rectangle with a length of 10 and a height of 9 has two right triangles on each side of it with short leg lengths of 9.

Answers

we have that

[the area of the trapezoid]=[area of rectangle]+2*[area of triangle]
[the area of the trapezoid]=[10*9]+2*[9*9/2]
[the area of the trapezoid]=[90]+2*[40.5]= 171 units²

see the attached figure

the answer is 171 units²

171 I think I need help too......

6 identical toys 1.8kg. B) what is the weight of one toy? give your answer in grams. C) what is the weight of 4 toys? give your answer in kilograms.

Answers

B=300grams
C=1.2Kg or 1200grams

Total weight of six identical toys = 1.8 Kg

Therefore, the weight of one such toy = 1.8 kg/ 6 = 0.3 kg

Further, 1 kg = 1000 gms

Answer to Ques B-

Weight of one toy in grams = 0.3 * 1000 grams

= 300 grams

Answer to Ques C-

Weight of 4 toys in kilograms = 0.3 * 4 kilograms

= 1.2 kilograms

Hope this helps..!!

Thank you :)

What is three fourths multiplied by 24

Answers

the answer is 6 jsjdjej2jdjfbebwjxnenwix d did bc sjsixne

18. You can divide 24 by 4 to get 6 (1/4 of 24), then multiply 6 by 3 to get 18 (3/4 of 24). You can also multiply 24 by 3, then divide by 4.

Convert 30 gallons to liters using an equivalent rate

Answers

113.62 liters is 30 gallons using equivalent rate

30 gallons is 113 liters

Lalo has 1500 minutes per month on his cell phone plan. How many more minutes can he use if he has already talked for 785 minutes? What is the interpretation?

Answers

1500 - 785 = 715
number of minutes per month - number of minutes used = number of minutes left.

Find the fifth term of an=2(-1) n

Answers

The 5th answer in this series would be -2.

We know this because in any problem where we have a base (2) being multiplied by a -1^n power, the numbers will simply alternate between positive 2 and negative 2. All of the even powers will be positive 2 and all of the odd numbers will be negative 2. See the work below.

2(-1)(-1)(-1)(-1)(-1)
2 (1)(-1)(-1)(-1)
2(1)(1)(-1)
2(-1)
-2

So since 5 is odd, the answer is negative 2.

Answer:

The fifth term of [tex]a_n = 2(-1)^n[/tex] is, -2

Step-by-step explanation:

Given the sequence:

[tex]a_n = 2(-1)^n[/tex] .....[1]

where,

n is the number of terms.

To find the fifth term of the given sequence:

Substitute n = 5 in [1] we have;

[tex]a_5 = 2 \cdot (-1)^5[/tex]

⇒[tex]a_5 = 2 \cdot -1 = -2[/tex]

Therefore, the value of fifth term of [tex]a_n[/tex] is, -2

There are 203 apples in a basket. How many children can share these apples equally? How many apples would each child get?

Question#1: All possible numbers of children in ascending order are:

Question#2: All possible numbers of apples in descending order are:

please help!!!!!!!!!

Answers

Q.1 1,7,29,203
Q.2 203,29,7,1

Answer:

Question 1: 1, 7, 29, 203

Question 2: 203, 29, 7, 1

Step-by-step explanation:

Please help me with question 6

Answers

Your answer choices are cut off. Only A and B are showing. Both A and B are true, so they are not the answer. A is true because the midsegment is always half that of the parallel side. Choice B is true because of similar triangles leads to congruent corresponding angles which means the sides are parallel.

The answer is either C or D. Choice C seems to show up but only partially. Choice D seems to be completely left out for some reason. Please repost with the answer choices.

PLEASE HELP I WILL FAIL MATH I will give you brainest

Answers

A- Part
B- $8
You can do 32 x .25 or 32/4

The difference between two numbers is 66. The sum of the two numbers is 88. Find the greatest number

Answers

It is going to be a greater number than 66. And a number less than 88. Hope this helps.

There is a spinner with 14 equal areas, numbered 1 through 14. If the spinner is spun one time, what is the probability that the result is a multiple of 5 or a multiple of 2?

Answers

The probability that a spinner with 14 equal areas lands on a multiple of 5 or a multiple of 2 is 4/7, after identifying the unique numbers from 1 to 14 that fit the criteria and dividing by the total number of possible outcomes.

The student is asking about the probability of an event occurring after spinning a spinner with 14 equal areas numbered 1 through 14. Specifically, the question seeks the probability that the result is either a multiple of 5 or a multiple of 2 when the spinner is spun once.

To solve this, first identify the multiples of 5 and 2 between 1 and 14: for 5, the multiples are 5 and 10; for 2, the multiples are 2, 4, 6, 8, 10, 12, and 14. Here, the number 10 is a multiple of both 5 and 2, so it should only be counted once. In total, there are 8 unique numbers that are multiples of 2 or 5 on the spinner. Since there are 14 possible outcomes, the probability is:

Probability = Number of favorable outcomes / Total number of outcomes = 8 / 14 = 4/7.

Thus, the probability that the spinner lands on a multiple of 5 or a multiple of 2 is 4/7.

QUICKK EXPERTS/ACE/GENIUSES

Answers

The answer is C because a translation is just moving the object up, down, left, or right. In the other three images, the picture is either reflected or rotated, which is not a translation.

the correct answer would be C.

the distance around a circle (the circumference) is about 3.14 times the diameter. If a circulares table has a diameter of 3 feet What is the circumference can somebody please help me I'm stuck xD

Answers

Well.
The relation between the circumference of a circle and its diameter is described by the following equation: [tex] Thecircumference=\pi d[/tex]
where d is the diameter of the circle.
For a giving diameter (d=3ft), we can substitute in the previous equation
The circumference of a circule= 3.14*3 (you can use ur calculator to find the exact number i guess it will be about 9.42 ft)

I hope it helps.

Japan accounts for about 5.4% of the world's petroleum consumption. Write this percent as a fraction.

Answers

The answer is: " [tex] \frac{27}{500} [/tex] " .
________________________________________________________

5.4 % = 5.4 / 100 = 54/1000 = (54÷2) / (1000÷2) = " [tex] \frac{27}{500} [/tex] " .

OR:

5.4 % = 5.4 / 100 = 5.4 ÷ 100 = 0.054 = 54/1000 = (54÷2) / (1000÷2) =

" [tex] \frac{27}{500} [/tex] " .
________________________________________________________

"The correct fraction representing Japan's petroleum consumption as a percentage of the world's total is [tex]$\frac{5.4}{100}$.[/tex]

To express the given percentage as a fraction, one must divide the percentage by 100, since percentages are parts per hundred. T

hus, 5.4% can be written as a fraction by placing 5.4 over 100, which gives us [tex]$\frac{5.4}{100}$[/tex].

Finally, this fraction can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case, yielding the simplified fraction [tex]$\frac{27}{500}$.[/tex]

When reduced to a fraction without a decimal, [tex]$\frac{27}{500}$."[/tex]

True or false ? The polynomial below is a perfect square trinomial of the form A2-2AB+B2 4x2-16x+16

Answers

True

4x^2 -16x +16 = (2x -4)^2

A = 2x
B = 4

How many triangles are in a dodecagon

Answers

A dodecagon has 12 triangles

Which of the polygons listed below have at least three angles?

I Triangles
II Quadrilaterals
III Pentagons
IV Hexagons
A. III and IV
B. II, III, and IV
C. I, II, III, and IV
D. IV

Answers

C: All of the above; I, II, III, and IV

it's C they all have at least three angles

(−6)∙(a+2b−3c−4d)−(−2)∙(−4a−3b+2c+d)

Answers

(−6)∙(a+2b−3c−4d)−(−2)∙(−4a−3b+2c+d)
this will give us:
-6(a+2b-3c-4d)+2(-4a-3b+2c+d)
opening the parenthesis we get:
-6a-12b+18c+24d-8a-6b+4c+2d
putting like terms together we get
-6a-8a-12b-6b+18c+4c+24d+2d
simplifying the above we get
-14a-18b+22c+26d
the answer is :
-14a-18b+22c+26d

Answer:

-14a-18b+22c+26d

a collection of dimes and nickels is worth $8.40 there are 106 coins how many of each are there

Answers

There are 62 dimes and 44 nickels.

N + D = 106 (There are 106 coins in all.)
1 nickel = 5 cents.
1 dime = 10 cents.
5*N + 10*D = 840 ($8.40 = 840 cents)
N + D = 106 -------(1)
N = (106 - D)
5*N + 10*D = 840 ---------(2)
5*(106-D) + 10*D = 840
530 - 5*D + 10*D = 840
5*D = 840 - 530
5*D = 310
D = 310/5 = 62
Eq(1) N + D = 106
N + 62 = 106
N = 106 - 62
N = 44

Use the distributive property to simplify the expression. 8(3 + 4) = 24 +

Answers

Using the distributive property is done by multiplying the factor (outside the parentheses) to each term within it. In this case, 8(3 + 4), we multiply the 8 by 3, and then add it to the product of 8 x 4:
8(3 + 4) = 8 x 3 + 8 x 4 = 24 + 32
If we need to simplify further, this adds up to 56.

The answer is 8(3 + 4) = 24 + 32

evaluate if hj + h if h = 6 and j = 8

Answers

The answer to your problem is 54.

6 multiplied by 8 plus 6-
8 multiplied by 6 is 48
48 plus 6 is 54

NEED HELP!!!!!!!!!!!!!!!!!!!!FAST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! kind like..........RIGHT
NOW!!!!!!!!!!!!!!!!!!!!!!!!!
The area of a circle is 113.04 in. 2. What is the radius of the circle? 36 inches 18 inches 12 inches 6 inches

Answers

the formula for the radius is

[tex] \sqrt{ \frac{a}{\pi} } [/tex]
so the answer would be 6 inches

the answer to this question would be 6 inches

given the input in the table which function rule produces the output?

Answers

Try this option:
4*4+2=18; 5*4+2=22; 7*4+2=30; 8*4+2=34
Answer: y=4x+2

Answer: b) [tex]f(x)=4x+2[/tex]

Step-by-step explanation:

The given table :-

Input 4 5 7 8

Output 18 22 30 34

Let's check all the options

a) [tex]f(x)=5x-2[/tex]

At x= 4, [tex]f(4)=5(4)-2=18[/tex]

At x= 5 , [tex]f(5)=5(5)-2=25-2=23\neq22[/tex] , thus it not the correct function.

b) [tex]f(x)=4x+2[/tex]

At x= 4 , [tex]f(4)=4(4)+2=18[/tex]

At x= 5 , [tex]f(5)=4(5)+2=22[/tex]

At x= 7 , [tex]f(7)=4(7)+2=30[/tex]

At x= 8 , [tex]f(8)=4(8)+2=34[/tex] , thus it is the function rule that produces the output.

c) [tex]f(x)=3x+6[/tex]

At x= 4 , [tex]f(4)=3(4)+6=18[/tex]

At x= 5 , [tex]f(5)=3(5)+6=21\neq22[/tex] , thus it not the correct function.

d) [tex]f(x)=2x+10[/tex]

At x= 4 , [tex]f(4)=2(4)+10=18[/tex]

At x= 5 , [tex]f(5)=2(5)+10=20\neq22[/tex] , thus it not the correct function.

Hence, the correct function rule that produces the output = b) [tex]f(x)=4x+2[/tex]

a cylinder shaped drum is used as a garbage container. The drum has a height of 4 ft and a radius of 1.25 ft how many cubic feet of garbage does the drum hold ? enter your answer as a decimal rounded to the nearest hundreth

Answers

To solve this problem you must apply the formula for calculate the volume of a cylinder, which is shown below:

V=πr²h

V is the volume of the cylinder.
r is the radius of the cylinder (r=1.25 ft).
h is the height of the cylinder (h=4 ft).

When you substitute these values into the formula ofr calculate the volume of the cylinder, you obtain:

V=πr²h
V=π(1.25 ft)²(4 ft)
V=19.63 ft³

Therefore, the answer is: 19.63 ft³

Answer:

19.63

Step-by-step explanation:

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