diff --git a/src/basic/phi-function.txt b/src/basic/phi-function.txt
index 52ec02ef0..3dfc7420b 100644
--- a/src/basic/phi-function.txt
+++ b/src/basic/phi-function.txt
@@ -2,32 +2,70 @@
#Euler's Totient Function
Also known as $\phi (n)$ i.e. **phi-function** it is just a count of coprimes (numbers,
-having greatest common divisor of `1`) with `n` and not exceeding `n` itself.
-Here `1` is regarded as a coprime to any number.
+having greatest common divisor of `1`) with $n$ and not exceeding $n$ itself.
+Here $1$ is regarded as a coprime to any number.
-For example, `4` has two such coprimes - `1` and `2`. Any prime, `n` (like `7`) obviously have
-`n-1` such coprimes.
+$\phi (n) =$ the number of positive integers less than $n$ that are relatively prime to $n$
+where $n >=1$.
+
+For example, $4$ has two such coprimes - $1$ and $2$. Any prime, $n$ (like $7$) obviously have
+$n - 1$ such coprimes.
Here are values of $\phi(n)$ for first few positive integers:
- N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
+ N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
- Phi 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 12 10
+ Phi(N) 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 12 10
+
+###Properties
+
+The following properties of Euler Totient Functin are enough to calculate if for any number.
+
+* If $p$ is a prime number, $\phi (p) = p - 1$ as $gcd(p, q) = 1$ where $1 <= q < p$.
+* If $p$ is a prime number and $k > 0$, $\phi (p^k) = p^k - p^{(k - 1)}$
+* If $a$ and $b$ are relatively prime, $\phi (ab) = \phi (a) . \phi (b)$
+
+We can get easily Euler function for any $n$ through factorization ( decomposing $n$ into its prime factors ).
+If
+>$n = {P_{1}}^{a_{1}} \cdot {P_{2}}^{a_{2}} \cdot {P_{3}}^{a_{3}} \cdot \ldots \cdot {P_{k}}^{a_{k}}$
+where $P_{i}$ are prime factors of $n$.
+
+>$\phi (n) = \phi ({P_{1}}^{a_{1}}) \cdot \phi ({P_{2}}^{a_{2}}) \cdot \ldots \cdot \phi ({P_{k}}^{a_{k}})$
+$= ({P_{1}}^{a_{1}} - {P_{1}}^{a_{1} - 1}) \cdot ({P_{2}}^{a_{2}} - {P_{2}}^{a_{2} - 1}) \cdot \ldots \cdot ({P_{k}}^{a_{k}} - {P_{k}}^{a_{k} - 1})$
+$= n \cdot (1 - \frac{1}{{P_{1}}^{a_{1}}}) \cdot (1 - \frac{1}{{P_{2}}^{a_{2}}}) \cdot \ldots \cdot (1 - \frac{1}{{P_{k}}^{a_{k}}})$
###Algorithm
-Here is the code for generating `phi(n)` in `Python 3`:
-
- def phi(n):
- result = n
- i = 2
- while i * i <= n:
- if n % i == 0:
- while n % i == 0:
- n //= i
- result -= result // i
- i += 1
- if n > 1:
- result -= result // n
- return result
+'C++' implementation:
+
+ int phi(int n) {
+ int result = n;
+ for(int i = 2; i * i <= n; ++i)
+ if(n % i == 0) {
+ while(n % i == 0)
+ n /= i;
+ result -= result / i;
+ }
+ if(n > 1)
+ result -= result / n;
+ return result;
+ }
+
+###Applications of Euler's function
+
+The most famous and important property of Euler's function is expressed in **Euler's theorem** :
+$a^{\phi (m)} = 1 (mod$ $m)$ $where$ $a$ and $m$ are relatively prime.
+
+In the particular case when, $m$ is a prime Euler's theorem turns into the so-called **Fermat's little theorem** :
+$a^{(m - 1)} \equiv 1 \pmod m$
+Euler's theorem occurs quite often in practical applications
+
+###Practice Problems
+
+* [SPOJ #4141 [Euler Totient Function] [Difficulty: CakeWalk]](http://www.spoj.com/problems/ETF/)
+* [UVA #10179 "Irreducible Basic Fractions" [Difficulty: Easy]](http://uva.onlinejudge.org/index.php?option=onlinejudge&page=show_problem&problem=1120)
+* [UVA #10299 "Relatives" [Difficulty: Easy]](http://uva.onlinejudge.org/index.php?option=onlinejudge&page=show_problem&problem=1240)
+* [UVA #11327 "Enumerating Rational Numbers" [Difficulty: Medium]](http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=2302)
+* [TIMUS #1673 "tolerance for the exam" [Difficulty: High]](http://acm.timus.ru/problem.aspx?space=1&num=1673)
+**We will add more practice problems soon**
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