# 2 Problems

• Mar 22nd 2009, 06:36 AM
Jimmy_W
2 Problems
Calculate $\displaystyle \varphi (n)$ for $\displaystyle n = 21^{21}$ and $\displaystyle n=2009$ In each case, express the answer as the product of prime numbers

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

thanks
• Mar 23rd 2009, 02:30 AM
CaptainBlack
Quote:

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

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

thanks

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

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

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

CB