No Relation?

The question asks for the sum of co-primes to 360 which are less than it :

Fortunately there exists a formula for it ;)

$\Rightarrow$ The sum of co-primes of N which are less than it is given by : $\dfrac{N}{2}\cdot \phi(N)$

Where $\phi(N)$ is the number of co-primes to N which are less than it .

$$ Now $\phi(N)$ can be calculated as follows :

• First we express N as a product of it's Prime Factors .

$N=a^{p}\times b^{q} \times c^{r} \dots$ (say)

• $\phi(N) = N\cdot \left ( 1 - \frac{1}{a} \right ) \cdot \left ( 1 - \frac{1}{b} \right ) \cdot \left ( 1 - \frac{1}{c} \right ) \dots$

$$ Now the N in this question is 360 .

$360=2^{3} \times 3^{2} \times 5$

Hence , s(360) = $\dfrac{N}{2}\cdot \phi(N) \\= \dfrac{360}{2} \cdot 360\cdot \left ( 1 - \frac{1}{2} \right ) \cdot \left ( 1 - \frac{1}{3} \right ) \cdot \left ( 1 - \frac{1}{5} \right ) \\= 180\times 96 \\= 17280$