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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. _________________________________________ 2. [CSIR 2012]For a positive integer m, let φ(m) denote the number of integers k such t...