Question

**Determine ?(????) using Euler’s totient
function**

Answer #1

**Totient** **Funtion** : Let us
consider a number N. Totient function gives the number of positive
integers that are less than N that are co-primes to N.

Here P_{1},P_{2},P_{3},.....
P_{k} are the prime factors of N

Given N=2700.

2700= 2^{2}.3^{3}.5^{2}

= 720.

So there are 720 number of positive integers that are less than 2700 which are co-primes of 2700 is present.

I hope I have clearly explained the procedure with calculation. If you want you can try with other examples as well. If you find any difficulties please feel to comment. I will be available to solve your quires. Please support by upvote if you like my presentation and explanation.

Prove Euler’s theorem: if n and a are positive integers with
gcd(a,n)=1, then aφ(n)≡1 modn, where φ(n) is the Euler’s function
of n.

Using Euler’s formula, show that for any natural number n,
cos(nθ) + isin(nθ) = (cos(θ) + isin(θ))n Using this, show that
cos(3θ) = cos3 (θ) − 3 cos(θ) sin2 (θ).

Determine the convergence or divergence if each integral by
using a comparison function. Show work using the steps below:
A. Indicate the comparison function you are using.
B. Indicate if your comparison function is larger or smaller
than the original function.
C. Indicate if your comparison integral converges or diverges.
Explain why.
D. State if the original integral converges or diverges. If it
converges, you don’t need to give the value it converges to.
11. integral from 1 to infinity...

Use Euler’s Theorem to find the inverse of 5 (mod 462).

Use Euler’s Method to obtain a five-decimal approximation of the
indicated value. Carry out the recursion by hand, using h = 0.1 and
then using h = 0.05.
y′ = -y + x + 1, y(0) = 1. Find
y(1)

please answer the question:
Describe the basic idea behind Euler’s Method. Compare this with
the Improved Euler’s Method - in
what way is it an improvement? Finally, compare both these methods
with the Runge-Kutta method -
what is the difference, and why does Runge-Kutta give more accurate
results?

explain and derive...
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Equations of motion of spring mass systems using Lagrangian
Equations of motion of pendulum spring etc using Newtons law
Gyroscope moment of inertia and angular velocity
Angular velocity of spools spinning
Angular momentum of disks rotating
Spring- mass systems, particularly spring attached to heavy
disk
Impacts and angular velocities after impact

Euler's Totient Function
Let f(n) denote Euler's totient function; thus, for a positive
integer n, f(n) is the number of integers less than n which are
coprime to n. For a prime p its is known that f(p^k) = p^k-p^{k-1}.
For example f(27) = f(3^3) = 3^3 - 3^2 = (3^2) 2=18. In addition,
it is known that f(n) is multiplicative in the sense that
f(ab) = f(a)f(b)
whenever a and b are coprime. Lastly, one has the celebrated
generalization...

For the initial value problem, Use Euler’s method with a step
size of h=0.25 to find approximate solution at x = 1

1. Consider the initial value problem dy/dx =3cos(x^2) with
y(0)=2.
(a) Use two steps of Euler’s method with h=0.5 to approximate
the value of y(0.5), y(1) to 4 decimal places.
b) Use four steps of Euler’s method with h=0.25, to
approximate the value of y(0.25),y(0.75),y(1), to 4 decimal places.
(c) What is the difference between the two results of Euler’s
method, to two decimal places?

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