1. ## 2 Problems

Calculate $\varphi (n)$ for $n = 21^{21}$ and $n=2009$ In each case, express the answer as the product of prime numbers

Let a,b be natural numbers. Show that if $a|b$ then $\varphi (a)| \varphi (b)$

thanks

2. Originally Posted by Jimmy_W
Calculate $\varphi (n)$ for $n = 21^{21}$ and $n=2009$ In each case, express the answer as the product of prime numbers

Let a,b be natural numbers. Show that if $a|b$ then $\varphi (a)| \varphi (b)$

thanks
Here we assume that $\varphi(n)$ denotes Euler's Totient function. In which case to answer this question you need the result that:

$
\varphi(n)= (p_1-1)p_1^{k_1-1}(p_2-1)p_2^{k_2-1} .. (p_r-1)p_r^{k_r-1}
$

where $n=p_1^{k_1} p_2^{k_2} .. p_r^{k_r}$ is the prime decomposition of $n$.

CB