Euler's theorem MY POINT OF VIEW

THANKING YOU ALL MY GODS.

THANKING YOU LORD SHIVA FOR GIVING SUCH WONDERFULLIFE.

THANKING YOU MOM AND DAD.

                                                       OM JAI SHRI RAMA

This article is about Euler's theorem in number theory. For other uses, see List of topics named after Leonhard Euler.

In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if n and a are coprime positive integers, then

${\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}}$

where ${\displaystyle \varphi (n)}$ is Euler's totient function. (The notation is explained in the article modular arithmetic.) In 1736, Leonhard Euler published his proof of Fermat's little theorem,^[1] which Fermat had presented without proof. Subsequently, Euler presented other proofs of the theorem, culminating with "Euler's theorem" in his paper of 1763, in which he attempted to find the smallest exponent for which Fermat's little theorem was always true.^[2]

The converse of Euler's theorem is also true: if the above congruence is true, then ${\displaystyle a}$ and ${\displaystyle n}$ must be coprime.

The theorem is a generalization of Fermat's little theorem, and is further generalized by Carmichael's theorem.

The theorem may be used to easily reduce large powers modulo ${\displaystyle n}$. For example, consider finding the ones place decimal digit of ${\displaystyle 7^{222}}$, i.e. ${\displaystyle 7^{222}{\pmod {10}}}$. Note that 7 and 10 are coprime, and ${\displaystyle \varphi (10)=4}$. So Euler's theorem yields ${\displaystyle 7^{4}\equiv 1{\pmod {10}}}$, and we get ${\displaystyle 7^{222}\equiv 7^{4\times 55+2}\equiv (7^{4})^{55}\times 7^{2}\equiv 1^{55}\times 7^{2}\equiv 49\equiv 9{\pmod {10}}}$.

In general, when reducing a power of ${\displaystyle a}$ modulo ${\displaystyle n}$ (where ${\displaystyle a}$ and ${\displaystyle n}$ are coprime), one needs to work modulo ${\displaystyle \varphi (n)}$ in the exponent of ${\displaystyle a}$:

if ${\displaystyle x\equiv y{\pmod {\varphi (n)}}}$, then ${\displaystyle a^{x}\equiv a^{y}{\pmod {n}}}$.

Euler's theorem is sometimes cited as forming the basis of the RSA encryption system, however it is insufficient (and unnecessary) to use Euler's theorem to certify the validity of RSA encryption. In RSA, the net result of first encrypting a plaintext message, then later decrypting it, amounts to exponentiating a large input number by ${\displaystyle k\varphi (n)+1}$, for some positive integer ${\displaystyle k}$. In the case that the original number is relatively prime to ${\displaystyle n}$, Euler's theorem then guarantees that the decrypted output number is equal to the original input number, giving back the plaintext. However, because ${\displaystyle n}$ is a product of two distinct primes, ${\displaystyle p}$ and ${\displaystyle q}$, when the number encrypted is a multiple of ${\displaystyle p}$ or ${\displaystyle q}$, Euler's theorem does not apply and it is necessary to use the uniqueness provision of the Chinese Remainder Theorem. The Chinese Remainder Theorem also suffices in the case where the number is relatively prime to ${\displaystyle n}$, and so Euler's theorem is neither sufficient nor necessary.

Indian politics,ruller and opposition party

First of all I bid namaste to my dad my mom and lord rameshwara..

I bid namaste to all seniors and gurus..

I thank you to all my good friends well wisher,followers and lovers.

With respect I always bow down to thee who are good soft and honest less speaking persons,active and believe in action.

The person who are really good cannot go to politics and will not be so confident in politics because everyone knows politics mean illusion to common, something which in based on lies and lots of shade not clear..

The person who are really not good and can achieve success in politics as like marketing agents and shopkeeper.

If the marketing agent speech right words than it will very difficult to be successful,like that if a political leaders says right speech fulfil all promise given to the common,than he can't be a successful leaders.

The person who can divert the common people mind to the different path can be successful in politics.

Like prime minister of India and the leaders of Indian NDA govt didn't do anything in the time of tennure.

NDA govt didn't complete any wish to the common people, like fulfil of the black money, return of black money, reduce the corruption, reductions of common people harrassment, reduction the proverty.

Our NDA govt did not fulfill the job employment,the largest young unemployment is in our country.

No job in govt which hamper the young ang cheating to the young and youth boys.

Sealing and GST close thousands lakh private company,and out govt kill the young men,youth and our all persons who was in service of there.

The NDA govt can't have any plan for farmers, cultivation and the implementation of cultivation in India.

BJP govt don't know how to GDP can be grown otherwise this govt should not focus on digital India which passively effect to the cultivation.

There is no such educated ministry and back up in NDA govt who can focus on the GDP growth and common farmers and young can implement their life..

If you know dharma and Bhagwat Gita you must be aware of softness and the person who didn't study can't be a soft hearted and can't be a proper ruller.

Only difference between a educated and uneducated person is the uneducated must show more fake smartness, uneducated must be more talkative, uneducated must be more lier,educated must be rude,and show his dress.

This is a very great example of our politics you can observe.

The mobile phone and social media and other device is in the market which destroy the common people future and be more dangerous for the youth.Our govt provide a minimum range phone to all person to watch internet.it is very good politics ,if you bye phone than you must have to recharge.

In the past time people buy a clothes every month but now recharge in phone and give GST to govt..

At the end time NDA kill some terrorist and said it is the BJP who can believe in kill against killing.

There is no dharma and good ideas in their mind.

It is completely not acceptable to a govt where a attach of Jammu is the agenda of a govt.

It is just work of defence force,and we are proud that it can be done by a high rank soldiers to kill other.

Killing against killing is very simple vogus and unethical uneducated person thinking where humanity is being killed.

To save a young man life govt should provide job to the educated person despite the govt provide nomination to uneducated person.

A lot of example you can see in Indian rulling party and other party.

Thanking you so much.

 

Indian politics,good and bad

India is such country where people are so uneducated, unknown,unknowledge,that it is very simple to do politics..Just provide a wrong knowledge to them by media they will vote you...

BJP manifesto a bunch of lies: Congress

The Congress on Monday called the BJP manifesto for the 2019 Lok Sabha polls a bunch of lies and said the ruling party had not mentioned anything about jobs, goods and services tax (GST), black money and demonetisation.

Senior party leaders, including Ahmed Patel and Randeep Surjewala among others, also released a document that detailed 125 ‘failed promises’ by the BJP after five years in government.

The Congress rubbished the BJP’s claim of doubling farmers’ income and pointed out it would take 28 years to do so at the current rate of agricultural growth.

It said the government earmarked only five paisa per girl child under the ‘Beti Bachao Beti Padhao’ (Save the girl child programme) and claimed that the government debt during Mr. Modi’s tenure grew by ₹27 lakh crore

Mr. Surjewala said no BJP leader, whether it was Prime Minister Narendra Modi or party president Amit Shah, Ministers Rajnath Singh or Sushma Swaraj, spoke on these crucial issues.

“It is going lies versus nyay (justice). They should bring out a maafi nama(letter of apology)….People have now realised that their promises remain only on their website as they adopt gimmicks to stay in power. Sometimes, he [Mr. Modi] becomes chowkidaar (watchman), chaiwala, kaamdar (toiler), fakir (ascetic) but never fulfils his promises,” said Mr. Patel.

Mr. Surjewala questioned the BJP on the promises made in 2014 and said that it’s time for the BJP to wind up its government since people have seen through it.

“You promised two crore jobs every year, which translates to 10 crore jobs in five years. In reality, during these five years, the number of jobs has actually reduced by 4.7 crore. And it’s not something that I am saying, but what has been said by the government’s own NSSO (National Sample Survey Organisation) data,” he told reporters.

The Congress leader also said that the BJP had not brought back ₹80 lakh crore of black money and deposited ₹15 lakh in each citizen’s bank account as it had promised.

“The reality was that Lalit Modi, Nirav Modi, Mehul Choksi and Vijay Mallya looted ₹1 lakh crore of the taxpayer’s money and fled the country right under Mr. Modi’s nose.”

Transparency in the judiciary

suit standard be different for different arm of the states the issues of the transparency has been missing. For the highest judiciary.

ISRO successfully launched a satellite into Orbit

the Indian Space Research Organisation successfully launched and electronic intelligence satellite emeset for the defence research and development organisation along with its 28 third party satellite.

the notable aspect of the launches new orbit and variant of the polar satellite launched vehicle PSLV switching on the 4th Street engine couple of times in which of the fourth stage of the orbital platform cleaning the three experiment l payload.

PSLV first place this 436 KTM is set into the 749 kilometre Orbit.

Then the rocket was brought down to put the 28 satellite into Orbit at an altitude of 504 kilometre.

then the rocket was brought down further to 485 kilometre for the forest region transformed into the play load platform caring 3 experimental payloads.

Automatic identification system

For many time satellite application capturing message transmitting from ships.

automatic pocket repeating system from the AMS 80 radio Limited satellite Corporation India to assist this amateur radio operator in tracking the monitoring position data.