[tex]3^{1000}[/tex] is congruent to 80 (mod 100).
To compute[tex]3^{1000}[/tex] mod 100 by hand, we can use modular arithmetic.
First, we can break down 100 into its prime factors:[tex]100 = 2^2 \times 5^2.[/tex].
This means that we can compute [tex]3^{1000}[/tex] mod 100 by separately computing [tex]3^{1000}[/tex] mod [tex]2^2[/tex] and [tex]3^{1000}[/tex] mod 5^2.
To compute [tex]3^{1000}[/tex] mod [tex]2^2[/tex], we can use the fact that [tex]3^2 = 9[/tex] is congruent to 1 mod 4.
Therefore, we can write:
[tex]3^{1000}[/tex] mod [tex]2^2 = (3^2)^{500} mod 2^2 = 1^500 mod 2^2 = 1[/tex]
To compute 3^1000 mod 5^2, we can use Euler's totient theorem, which states that if a and n are coprime (i.e. their greatest common divisor is 1), then [tex]a^phi(n)[/tex] is congruent to 1 mod n,
where phi(n) is the Euler totient function.
Since 3 and 25 are coprime (their greatest common divisor is 1), we have:
[tex]\phi(25) = (5-1)\times (5) = 20[/tex]
Therefore, we can write:
[tex]3^{1000} mod 25 = 3^{(20\times 50)} \times 3^{10 } mod 25 = 1\times 3^{10} mod 25[/tex]
Now we just need to compute [tex]3^10[/tex] mod 25.
We can do this by repeatedly squaring and reducing mod 25:
[tex]3^2 = 9[/tex]
[tex]3^4 = 81 = 6 mod 25[/tex]
[tex]3^8 = 36^2 = 11^2 = 121 = 21 mod 25[/tex]
[tex]3^{10} = 3^8 \times 3^2 = 21\times 9 = 189 = 14 mod 25[/tex]
Therefore, we have:
[tex]3^{1000} mod 25 = 3^{10} mod 25 = 14[/tex]
Now we can use the Chinese remainder theorem to combine our results and find [tex]3^{1000}[/tex] mod 100.
Since [tex]2^2 and 5^2[/tex] are coprime (their greatest common divisor is 1), we can write:
[tex]3^{1000} mod 100 = (1\times25\times14 + 1\times4\times1) mod 100 = 1401 mod 100 = 1[/tex]
Therefore, [tex]3^{1000}[/tex] is congruent to 1 mod 100.
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Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.
x < 5
–6x – 5 < 10 – x
–6x + 15 < 10 – 5x
A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
Using the formula: A=p(1+r)t
How much money will Mario have if he puts $1000 in a savings account earning 4% annually after 10 years?
Select one:
O a. $1004
O b. $1428.24
• c. $1000
© d. $2156.01
The amount of money Mario will have if he puts $1000 in a savings account earning 4% annually after 10 years is $1,480
Compound interestA = p(1 + r)^t
p = $1000r = 4% = 0.04t = 10 yearsA = p(1 + r)^t
= 1000(1 + 0.04)^10
= 1000(1.04)^10
= 1000(1.48)
A = $1,480
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Apply the distributive property to factor out the greatest common factor 15+21
What is the term for breaking a larger number apart into smaller numbers that can be multiplied together to get a specific result?.
Distributive property is the term for breaking a larger number apart into smaller numbers that can be multiplied together to get a specific result.
What is distributive property?Firstly, to distribute means to divide something or give a share or part of something.
Distributive property explains that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.
Therefore, distributive property is the term for breaking a larger number apart into smaller numbers that can be multiplied together to get a specific result.
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As part of your retirement planning, you purchase an annuity that pays 5.75% annual interest compounded quarterly If you make quarterly payments of $1600 how much will you have saved in 5 years? Instead, if you make quarterly payments of $800, how much will you have saved in 10 years?
help me please im in a rush
Answer:
a.1
b. 17
Step-by-step explanation:
So the | | around an equation makes it an absolute value which can be defined as the "distance from zero" or in other words, if a value is positive, it stays positive, and if a value is negative, it becomes positive.
|-8 + 9| = |1| = 1
|-8| + 9 = 8 + 9 = 17
Based on the 2009 Kaiser Family Foundation survey, adding up the daily media use figures to obtain a report of weekly media use leads to the staggering number of more than _____ hours a week of media use by 11- to 14-year-olds.
The survey shows that adding up the daily media use figures to obtain a report of weekly media use leads to the staggering number of more than 60 hours a week of media use by 11- to 14-year-olds.
What is a survey?It should be noted that a survey simply means a list of questions that are used to get information about a group.
In this case, the daily media use figures to obtain a report of weekly media use leads to the staggering number of more than 60 hours a week of media use by 11- to 14-year-olds.
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Jordan is studying whether or not race affects high school graduation in utah. in this example, high school graduation is the?
The high school graduation is the dependent variable.
In this question,
Jordan is studying whether or not race affects high school graduation in utah.
A dependent variable is a variable in an expression that depends on the value of another variable. It is a variable that represents a quantity that changes based on other quantities being manipulated in an experiment. It is the variable being tested, and therefore, it is called the dependent variable.
In the above statement, the high school graduation gets affected by race. So, the the high school graduation is the dependent variable.
Hence we can conclude that the high school graduation is the dependent variable.
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Angles Related to a Circle. Solve for X
The value of x is (d) 1
How to solve for x?The angle ABC is given as:
∠ABC = 43x
The line AB touches the tangent line BC at point B.
However, the line AB is not a radius.
This means that:
∠ABC < 90
So, we have
43x < 90
Divide by 43
x < 2.09
From the list of options, the number less than 2.09 is 1
Hence, the value of x is (d) 1
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Which of the following is the complete factorization of the polynomial below?
2x² +13x² +17x-12
a
jfjoydx9tdzrzfzixitxxytxixutx
The function f(x) = 0.11(3)x is reflected over the x-axis to produce function g(x). Function g(x) is then reflected over the y-axis to produce function h(x). Which function represents h(x)?
h(x) = –0.11(3)x
h(x) = 0.11(3)–x
h(x) = 0.11(3)x
h(x) = –0.11(3)–x
Answer:
[tex]h(x)=-0.11(3)^{-x}[/tex]
Step-by-step explanation:
Given function:
[tex]f(x)=0.11(3)^x[/tex]
Reflection in the x-axis:
When reflecting a function in the x-axis, make the whole function negative:
[tex]\implies g(x)=-f(x)[/tex]
[tex]\implies g(x)=-0.11(3)^x[/tex]
Reflection in the y-axis:
When reflecting a function in the y-axis, make the x variable negative:
[tex]\implies h(x)=g(-x)[/tex]
[tex]\implies h(x)=-0.11(3)^{-x}[/tex]
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Answer:
the answer to this question is d
Step-by-step explanation:
h(x)=-0.11(3)^-x
PLEASE HELP DUE TONIGHT
1. The time taken for the water to fall that much is 0.2 s
2. The initial speed of the water is 4 m/s
1. How to determine the time Height (h) = 20 cm = 20 / 100 = 0.2 mAcceleration due to gravity (g) = 9.8 m/s²Time (t) =?h = ½gt²
0.2 = ½ × 9.8 × t²
0.2 = 4.9 × t²
Divide both side by 4.9
t² = 0.2 / 4.9
Take the square root of both side
t = √(0.2 / 4.9)
t = 0.2 s
2. How to determine the initial speedHorizontal distance (s) = 80 cm = 80 / 100 = 0.8 mTime (t) = 0.2 sInitial speed (u) =?s = ut
Divide both sides by t
u = s / t
u = 0.8 / 0.2
u = 4 m/s
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[tex](9x^{2} +9x-4)(9x^{2} +9x-10)-72[/tex]
Quadratic expressions are mathematical expressions that have the power of one of its alphabets being equal to and not more than two. The given expression in the question can be simplified as 81[tex]x^{4}[/tex] + 162[tex]x^{3}[/tex] - 45[tex]x^{2}[/tex] - 54x - 112
Algebraic expressions are mathematical expressions that consist of given number(s) and alphabet(s). When the power (exponential) of one of the alphabets is given as two, then the expression can be classified as a quadratic expression.
Thus the given expansion of the quadratic expression can be simplified as follows:
([tex]9x^{2}[/tex] + 9x - 4)([tex]9x^{2}[/tex] + 9x - 10) - 72
But,
([tex]9x^{2}[/tex] + 9x - 4)([tex]9x^{2}[/tex] + 9x - 10) = [tex]9x^{2}[/tex]([tex]9x^{2}[/tex] + 9x - 10) + 9x([tex]9x^{2}[/tex] + 9x - 10) - 4([tex]9x^{2}[/tex] + 9x - 10)
= 81[tex]x^{4}[/tex] + 81[tex]x^{3}[/tex] - 90[tex]x^{2}[/tex] + 81[tex]x^{3}[/tex] + 81 [tex]x^{2}[/tex] - 90x - 36 [tex]x^{2}[/tex] + 36x - 40
= 81[tex]x^{4}[/tex] + 162[tex]x^{3}[/tex] - 45[tex]x^{2}[/tex] - 54x - 40
So that we have;
(81[tex]x^{4}[/tex] + 162[tex]x^{3}[/tex] - 45[tex]x^{2}[/tex] - 54x - 40) - 72 = 81[tex]x^{4}[/tex] + 162[tex]x^{3}[/tex] - 45[tex]x^{2}[/tex] - 54x - 40 -72
= 81[tex]x^{4}[/tex] + 162[tex]x^{3}[/tex] - 45[tex]x^{2}[/tex] - 54x - 112
Thus the simplification of the given expression is 81[tex]x^{4}[/tex] + 162[tex]x^{3}[/tex] - 45[tex]x^{2}[/tex] - 54x - 112
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At the store, there were 6 bags of chips for every 8 cookie boxes. If there were 104 cookie boxes, how many bags of chips were there?
Answer: There were 78 bags of chips.
←
Which equation is represented by the graph below?
-5
oy-ex
40
3
2
-¹11
7 ?T?
3 45 x
The equation that is represented by the graph below is y = e^x - 4.
How to illustrate the graph?From the graph, it can be seen that the only exponential function n to hat intercept the function at equal to 3 is y = e^x - 4.
Let's equate y to 0.
y = e^x - 4.
e^x - 4 = 0
e^x = 4
In(e^x) = In(4)
x = 1.386
The x intercept is located at (1.386, 0).
In conclusion, the correct option is D.
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Find the vertex and axis of symmetry of each quadratic equation.
Answer: Vertex (-2,-15) and therefore the axis of sym will be -2
Step-by-step explanation: Using -b/2a you can deduce that 4x^2 is a, 16x is b and 1 is c. So -b/2a = -16/8 = -2. Then you plug -2 for y and yu should get -15. Then x will be your axis of symetry to x=-2
Which test represents the best choice if you wanted to compare the average mean time to repair (mttr) of your office for the last month to the division’s mttr?
The test that represents the best choice if you wanted to compare the average mean time to repair (mttr) of your office for the last month to the division’s mttr is one sample t-test
The purpose of the one sample t-test is to determine if a sample observations could follow a specific parameter like the mean.
This test is used to determine whether an unknown population mean is different from a specific value.
one sample t-test is used to compare the mean of a single group against that of a known mean.
We wanted to compare the average mean time to repair (mttr) of your office for the last month to the division’s mttr
So this test (one sample t-test ) would be the best choice to compare mttr.
Therefore, the test that represents the best choice if you wanted to compare the average mean time to repair (mttr) of your office for the last month to the division’s mttr is one sample t-test
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Find the area of the convex polygon with vertices (0,5), (-1,2), (4,4), (-3,-4) and (2,0).
The area of the convex polygon is 21.5 square units
How to determine the area of the convex polygon?The vertices are given as:
(0,5), (-1,2), (4,4), (-3,-4) and (2,0)
The area is then calculated as:
A = 0.5 * |x1y2 - x2y1 +x2y3 - x3y2 + ....... |
So, we have:
A = 0.5 * |-1 * 5 - 2 * 0 + 0 * 4 - 5 * 4 + 4 * 0 - 4 * 2 +2 * -4 - 0 * -3 - 3 * 2 + 4 * 1|
Evaluate
A = 0.5 * |-43|
Remove the absolute bracket
A = 0.5 * 43
This gives
A = 21.5
Hence, the area of the convex polygon is 21.5 square units
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please answer quickly
1a. 10. 24
1b. 125/243
2a. 0. 0048
2b. 32/3125
3a. 0. 64
3b. 0. 0031
4a. 2/5
4b. 48. 735
5a. 2. 36
5b. 1. 39
How to determine the values
1a. Given the values
(2/5)^2/(1/2)^6
Multiply both the numerator and denominator by the powers
⇒ [tex]\frac{\frac{4}{25} }{\frac{1}{64} }[/tex]
To find the common ration, multiply thus;
⇒ [tex]\frac{4}{25}[/tex] × [tex]\frac{64}{1}[/tex]
⇒ [tex]\frac{256}{25}[/tex]
= 10. 24
1b. (5/7)^2 × (5/7)^1
= [tex]\frac{25}{49}[/tex] × [tex]\frac{5}{7}[/tex]
= [tex]\frac{125}{343}[/tex]
= 125/243
2a. 0. 6^1 × 0. 2^3
= 0. 6 × 0. 008
= 0. 0048
2b. (2/5)^3 × (2/5)^2
= [tex]\frac{8}{125}[/tex] × [tex]\frac{4}{25}[/tex]
= [tex]\frac{32}{3125}[/tex]
= 32/ 3125
3a. 1^99 - 0. 6^2
= 1 - 0. 36
= 0. 64
3b. (0. 2 ) ^1 × (1/8)^2
= 0. 2 × 1/64
= 0. 2 × 0. 016
= 0. 0031
4a. (1/2)^2/ (5/8)^1
= [tex]\frac{\frac{1}{4} }{\frac{5}{8} }[/tex]
Take the inverse of the denominator
= [tex]\frac{1}{4}[/tex] × [tex]\frac{8}{5}[/tex]
= 2/5
4b. 7^2 - 0. 5^3
= 49 - 0. 125
= 48. 875
5a. 3^1 - 0. 8 ^2
= 3 - 0. 64
= 2. 36
5b. 0. 7^2 + 0. 9^1
= 0. 49 + 0. 9
= 1. 39
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1. Which fraction is equivalent to 2/3.¸?
(A) 4
9
B
C
(D)
0100
24
Answer:
The answer options appear garbled. If A is 4/9, then this is equal to 2/3. The others I can't see/understand.
Step-by-step explanation:
Fatima wants to find the value of sine theta, given cotangent theta = four-sevenths. which identity would be best for fatima to use? cosine theta = startfraction 1 over secant theta endfraction sine squared theta cosine squared theta = 1 cosecant theta = startfraction r over y endfraction
Answer:
Step-by-step explanation:
From the right triangle with cot theta = 4/7 = (adjacent side / opposite side), Fatima can find the length of the hypotenuse using Pythagoras. This gives a value of 8.0823. So cos theta = 4 / 8.0823 = 0,49614,
Then she could find the values of sin theta using
sine squared theta + cos squared theta = 1.
Answer: its D or 1 + cotangent squared theta = cosecant squared theta
Step-by-step explanation:
Porfa urgente para hoy
Si triplicamos la cantidad de cada artículo, el subtotal será de 2 940 soles, el impuesto al consumo será de 529,20 soles y el total será de 3 469,20 soles.
¿Cual es el coste total y los impuestos por cambiar la cantidad de artículos comprados?
En esta pregunta tenemos un recibo con una lista de productos, cantidades, precios unitarios y precios de compra, además de un impuesto que se aplica al consumo por concepto de los productos comprados y el total, el cual refleja tanto el subtotal como el concepto por impuestos.
En cuanto a la compra de una determinada cantidad para un artículo, el coste total respectivo es igual a producto del número de artículos por el precio unitario.
El subtotal se determina mediante la suma de los costes totales asociados a cada artículo y luego se le recarga con el impuesto al consumo. Por último, el total de compra es la suma del subtotal y los pagos tributarios.
Si triplicamos el número de cada artículo comprado, tenemos los siguientes costes para cada artículo:
Chompas de dralom
12 / 4 × 280 = 840
Chompas de Orlon
6 / 2 × 100
300
Chompas de lana de alpaca
15 / 5 × 600
1 800
El subtotal es 2 940 soles y el impuesto al consumo es 18 / 100 × 2 940 = 529,20 soles. Por tanto, el total es 3 469,20 soles.
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In 2016, Alberta had about 4.2 million people.
Assuming they follow the same population
growth rate, it is predicted they will have 6.65
million people in 20 years. At what rate is the
province's population growing?
the population of province is growing at the rate of 2.91%
What is the formula of the rate of growth of a population?the formula of the rate of growth of a population =
[(future population- initial population/initial population)*100]/time
population in 2016 = 4.2 milllion = 4200000
Population predicted in 20 years = 6.65 million = 6650000
so, initial population = 4200000
future population = 6650000
rate of growth = {(6650000- 4200000/4200000)*100]/20
= 58.33/20
= 2.91%
the population is growing at the rate of 2.91%
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Zoe went out to her garden and cut 19 roses from the first rose bush, 28 roses from the second rose bush, 37 roses from the third rose bush, and 46 roses from the fourth rose bush. What kind of sequence is this?
Based on the given scenario, the kind of sequence explained is an arithmetic sequence
SequenceFirst bush = 19 rosesSecond bush = 28 rosesThird bush = 37 rosesFourth bush = 46First term, a = 19
Second term, a + d
28 = 19 + d
28 - 19 = d
9 = d
Common difference, d = 9Third term, a + 2d
= 19 + 2(9)
= 19 + 18
= 37
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Expand.
If necessary, combine like terms.
(x+3)(x-3)=(x+3)(x−3)=left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, equals
The expanded form of the expression (x+3)(x-3) is x² - 9
What is the expanded form of the expression?Given the expression;
(x+3)(x-3)
We multiply each term in the second bracket by each term in the first bracket
x(x-3) + 3(x-3)
x² - 3x + 3x - 9
x² - 9
Therefore, the expanded form of the expression (x+3)(x-3) is x² - 9.
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John noticed that the angle formed by the minute hand and hour hand on a standard 12-hour clock was 110 degrees when he left home after 6 p.m.; it was also 110 degrees when he returned before 7 p.m. that same night. If he left home for more than five minutes, for how many minutes was he away
Jhon was between 6:12 p.m and 6:52 p.m. So, Jhon left home for exact 40 minutes. Using the angle made by the hands in the clock, the required value is calculated.
How to find the angle between the minute hand and hour hand?The general formula for finding the angle between the minute hand and the hour hand is
M = [tex]\frac{2}{11}[/tex] (H × 30 ± θ)
Calculation:It is given that,
Jhon left home after 6 p.m with an angle of 110° formed by the minute hand and hour hand.
So,
M = [tex]\frac{2}{11}[/tex] (H × 30 - θ)
= [tex]\frac{2}{11}[/tex] (6 × 30 - 110)
= [tex]\frac{2}{11}[/tex] (180 - 110)
= [tex]\frac{2}{11}[/tex] × 70
= 140/11
So, he left after 6:12 p.m
Jhon returned home before 7 p.m with angle of 110° formed by the minute hand and hour hand.
M = [tex]\frac{2}{11}[/tex] (H × 30 + θ)
= [tex]\frac{2}{11}[/tex] (6 × 30 + 110)
= [tex]\frac{2}{11}[/tex] (180 + 110)
= [tex]\frac{2}{11}[/tex] × 290
= 580/11
So, he returned before 6:52 p.m
Then,
The time taken for returning home = 580/11 - 140/11 = 440/11 = 40 minutes.
Therefore, he took 40 minutes to return home.
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NEED HELP ASAP
Lucy pays $275 in advance on her account at the athletic club. Each time she uses the club, $5 is
deducted from the account. Write a linear function that gives the value remaining in her
account after x visits to the club. Find the value remaining in the account after 11 visits.
Answer:
Step-by-step explanation:
Can someone help solve both of these? ASAP
Question 4
[tex]\sin 30^{\circ}=\frac{y}{14\sqrt{3}}\\\\\frac{1}{2}=\frac{y}{14\sqrt{3}}\\\\\boxed{y=7\sqrt{3}}\\\\\\\\\cos 30^{\circ}=\frac{x}{14\sqrt{3}}\\\\\frac{\sqrt{3}}{2}=\frac{x}{14\sqrt{3}}\\\\\boxed{x=21}[/tex]
Question 5
[tex]\cos 45^{\circ}=\frac{x}{28}\\\\\frac{1}{\sqrt{2}}=\frac{x}{28}\\\\x=\frac{28}{\sqrt{2}}\\\\\boxed{x=14\sqrt{2}}\\\\\\\\\sin 45^{\circ}=\frac{y}{28}\\\\\frac{1}{\sqrt{2}}=\frac{y}{28}\\\\y=\frac{28}{\sqrt{2}}\\\\\boxed{y=14\sqrt{2}}[/tex]
1. what are the peaks of a periodic function?
a. the highest points of the graph
b. the lowest points of the graph
c. the extremes of the graph
d. the x-intercepts of the graph
Answer:
A. The Highest Points Of The Graph
Step-by-step explanation:
A nonperiodic function is one that generates a graph without a regular recurring pattern over a fixed time interval. A repeating section of a graph is called the period.
What is the range of function g if g (x) = f(x) + 3?
A. (3,00)
B. (-3, 3)
C. (-∞0, ∞0)
D. (-∞0, 3)