If $N$ is the integer $2^43^35^27$ find the smallest positive integer $m$ such that $x^m \equiv 1 \mod N$ for all integers coprime to $N$ If $N$ is the integer $2^43^35^27$ find the smallest positive integer $m$ such that $x^m \equiv 1 \mod N$ for all integers coprime to $N$.
I understand that I can do this for individual $N$ ($N= 2^4, 3^3, 5^2, 7$) and then take the LCM by Chinese Remainder Theorem. Using Fermat's Little Theorem doesn't give me the largest such $m$ for each choice of $N$. What's the relevant theorem/tool here?
EDIT (per suggestions): Applying the Euler phi-function gives us $\phi(2^4) = 8, \phi(3^3) = 3^2\times 2, \phi(5^2) = 5*4, \phi(7) = 6$. 
Why are these the smallest such $m$?
 A: $\color{Green}{\text{Lemma}}$: 


*

*For every odd prime number $p$; 
and for every positive integer $\alpha$;
the multiplicative group $\mathbb{Z}_{p^{\alpha}}^*$;
is 
a cyclic group of order 
$\phi(p^{\alpha})= (p-1)p^{\alpha-1}$.
In other words:  


$$ 
\big( 
\mathbb{Z}_{p^{\alpha}}^* 
\ , \times 
\big) 
\equiv
\big( 
\mathbb{Z}_{(p-1)p^{\alpha-1}} 
\ , + 
\big) 
. 
$$


*

*For $\color{Red}{p=2}$; 
and for every positive integer $\color{Red}{3 \leq \alpha}$;
the multiplicative group $\mathbb{Z}_{2^{\alpha}}^*$;
is 
the direct sum of $\mathbb{Z}_2$ and 
a cyclic group of order 
$\color{Red}{\dfrac{1}{2}}\phi(2^{\alpha})= \color{Red}{2^{\alpha-2}}$.
In other words:  


$$ 
\big( 
\mathbb{Z}_{2^{\alpha}}^* 
\ , \times 
\big) 
\equiv
\big( 
\mathbb{Z}_2 
\oplus 
\mathbb{Z}_{\color{Red}{2^{\alpha-2}}} 
\ , + 
\big) 
. 
$$


*

*The multiplicative group $\mathbb{Z}_{2^2}^*$; 
is 
a cyclic group of order $2$.
The multiplicative group $\mathbb{Z}_{2}^*$; 
is 
the trivial group.




$\color{Teal}{\text{Remark}}$:  
Let's define the function $\psi$ as follows:  


*

*For every odd prime number $p$; 
and for every positive integer $\alpha$;
$ 
\psi(p^{\alpha}) 
= 
\phi(p^{\alpha}) 
= 
(p-1)p^{\alpha-1}
. 
$

*For $\color{Red}{p=2}$; 
and for every positive integer $\color{Red}{3 \leq \alpha}$;
$ 
\psi(2^{\alpha}) 
= 
\color{Red}{\dfrac{1}{2}}\phi(2^{\alpha}) 
= 
\color{Red}{2^{\alpha-2}} 
. 
$ 

*$\psi(4)=\phi(4)=2.$

*$\psi(2)=\phi(2)=1.$

*$\psi(1)=\phi(1)=1.$



*

*If $n$ has 
the prime factorization 
$ 
n 
= 
\color{Red}{2^{\alpha_0}} 
p_1^{\alpha_1} 
p_2^{\alpha_2} 
... 
p_k^{\alpha_k} 
$ 
;
with 
$\alpha_0 \in \mathbb{N}_0=\mathbb{N} \cup \{ 0 \} $ 
; 
$\alpha_1 \in \mathbb{N}$ 
, 
$\alpha_1 \in \mathbb{N}$ 
, 
$...$ 
$\alpha_k \in \mathbb{N}$ 
; 
then let's define: 
$ 
\psi(n) 
= 
\text{lcm} 
\Big( 
\psi(2^{\alpha_0}); 
\psi(p_1^{\alpha_1}), 
\psi(p_2^{\alpha_2}), 
..., 
\psi(p_k^{\alpha_k}) 
\Big) 
$






One can easilly checks that
:  
$$\color{Teal}{\psi(n)} 
\color{Green} 
{
\text{is the least integer} 
\ 
m 
\ 
\\ 
\text{such that} 
\ 
x^m \overset{n}{\equiv} 1 
\ 
; 
\\ 
\text{for all integers} 
\ 
x 
\ 
\text{coprime to} 
\ 
n 
\ 
}.$$

In your case the answer is:  
$$ 
\psi(\color{Red}{2^4} . 3^3 . 5^2 . 7) 
= 
\text{lcm} 
\Big( 
\psi(\color{Red}{2^4}); 
\ 
\psi(3^3), 
\ 
\psi(5^2), 
\ 
\psi(7) 
\Big) 
= 
\\ 
\ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ 
\text{lcm} 
\Big( 
\color{Red}{2^2};  
2 \times 3^2, 
4 \times 5, 
6 
\Big) 
= 
2^2 \times 3^2 \times 5 = 180 
. 
$$
