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Try this problem from TIFR GS-2010 which involves the concept of Cyclic Group.

## CYCLIC GROUP | TIFR 201O | PART A | PROBLEM 1

A cyclic group of order $60$ has

• $12$ generators
• $15$ generators
• $16$ generators
• $20$ generators

### Key Concepts

CYCLIC GROUP

ORDER OF AN ELEMENT

EULER’S PHI FUNCTION

But try the problem first…

Answer:$16$ generators

Source

TIFR 2010|PART A |PROBLEM 1

CONTEMPORARY ABSTRACT ALGEBRA: JOSEPH GALLIAN

## Try with Hints

First hint

If G be a cyclic group of order n ,then the number of generators of G is $\phi(n)$

Second Hint

Let us define Euler’s phi function:-

$\phi(n)$=the number of positive integers less than $n$ and prime to $n$

i)If $n$ is prime then $\phi(n)=n-1$

ii)If $m$,$n$ be two integers which are relatively prime $\phi{mn}=\phi(m)\phi(n)$

iii)If $n$ be a prime and $k$ be any positive integer,$\phi(p^k)=p^k(1-\frac{1}{p})$

Final Step

Now $60$ can be written as product of $2^2$,$3$,$5$

Therefore $\phi(60)=\phi(2^2).\phi(3).\phi(5)$

Now $\phi(2^2)=2$…….i

Now $\phi(3)=(3-1)=2$ ……..ii

Now $\phi(5)=(5-1)=4$ ……..iii

Now multiply i,ii &iii and get the result