Euler's Theorem states that if $gcd(a,n) = 1$, then $a^{φ(n)} ≡ 1 (mod \,n)$.
Here $φ(n)$ denotes the Euler Totient function. This theorem has many interesting applications. One of them is finding last digit or last two digits of a number, given in some power notation.
For example consider the following csir problem.
1. [CSIR 2012]The last two digits of $7^{81}$ are
a. 07. b. 17. c.37. d.47.
How to use Euler's Theorem here?
Last two digits of a number is nothing but congruent modulo 100. So our problem is nothing but to find $x$, where $x$ is given by
$7^{81}≡ x (mod \,100).$
By Euler's theorem we have $7^{16}≡ 1 (mod \,100).$ So, taking 5th power both sides $7^{80}≡ 1 (mod \,100).$
Now can you find the required answer?
In the same CSIR 2012 question paper in part C also, another question is asked based on Euler's theorem.
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2. [CSIR 2012]For a positive integer m, let φ(m) denote the number of integers k such that $1 \le k \lneq m$ and $GCD(k, m)=1$. Then which of the following statements are necessarily true?
a. ${φ(n)}$ divides $n$ for every positive integer $n$.
b. $n$ divides $a^{φ(n)} - 1$ for all positive integers $a$ and $n$.
c.$n$ divides $a^{φ(n)} - 1$ for all positive integers $a$ and $n$ such that GCD(a, n)=l.
d. $n$ divides $a^{φ(n)} - 1$ for all positive integers $a$ and $n$ such that GCD(a, n)=l.
__________________________________________
Similarly, we can find last digit also, by considering modulo 10.
3. [CSIR 2013] What is the last digit of $7^{73}$ ?
a. 7 b. 9 c. 3 d. 1
We welcome the reader, to find the answers.
Here $φ(n)$ denotes the Euler Totient function. This theorem has many interesting applications. One of them is finding last digit or last two digits of a number, given in some power notation.
For example consider the following csir problem.
1. [CSIR 2012]The last two digits of $7^{81}$ are
a. 07. b. 17. c.37. d.47.
How to use Euler's Theorem here?
Last two digits of a number is nothing but congruent modulo 100. So our problem is nothing but to find $x$, where $x$ is given by
$7^{81}≡ x (mod \,100).$
By Euler's theorem we have $7^{16}≡ 1 (mod \,100).$ So, taking 5th power both sides $7^{80}≡ 1 (mod \,100).$
Now can you find the required answer?
In the same CSIR 2012 question paper in part C also, another question is asked based on Euler's theorem.
_________________________________________
2. [CSIR 2012]For a positive integer m, let φ(m) denote the number of integers k such that $1 \le k \lneq m$ and $GCD(k, m)=1$. Then which of the following statements are necessarily true?
a. ${φ(n)}$ divides $n$ for every positive integer $n$.
b. $n$ divides $a^{φ(n)} - 1$ for all positive integers $a$ and $n$.
c.$n$ divides $a^{φ(n)} - 1$ for all positive integers $a$ and $n$ such that GCD(a, n)=l.
d. $n$ divides $a^{φ(n)} - 1$ for all positive integers $a$ and $n$ such that GCD(a, n)=l.
__________________________________________
Similarly, we can find last digit also, by considering modulo 10.
3. [CSIR 2013] What is the last digit of $7^{73}$ ?
a. 7 b. 9 c. 3 d. 1
We welcome the reader, to find the answers.
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