A Totient Quotient

Relevant wiki: Euler's Totient Function

Since 7 and 1000000 are coprime, $\phi(7)\phi(1000000) = \phi(7000000)$ .

So, $\frac{\phi(7000000)}{\phi(1000000)} = \phi(7)$ .

And, since 7 is prime, $\phi(7) = 7-1 = \boxed{6}$ .

We have to solve for: $\dfrac{\phi 7000000}{\phi 1000000}$

Using the formula for $\phi (n)$ we can expand it: $\large = \dfrac{ (7000000) (1 - \dfrac{1}{2} ) (1 - \dfrac{1}{5} ) (1 - \dfrac{1}{7} )}{ (1000000) (1 - \dfrac{1}{2} )(1 - \dfrac{1}{5})}$

After simplifying:

$\large = (7) (1 - \dfrac{1}{7}) = 7 \times \dfrac{6}{7} = \color{#3D99F6} \boxed{ 6 }$

$\displaystyle \frac { \phi ( 7000000 ) } { \phi ( 1000000 ) } = \phi ( 7 ) = 1, 2, 3, 4, 5, 6, \cancel { 7 } \Rightarrow \boxed { 6 } \text { 1 is a factor of every natural number. }$

$\large\phi\left (\cdot\right )$ is multiplicative, so: $\large\begin{aligned} \cfrac{\phi\left (7000000\right )}{\phi\left (1000000\right )}&=\phi\left (\cfrac{7000000}{1000000}\right )\\ &=\phi\left (7\right )\\ &=7\left(1-\cfrac{1}{7}\right)\\ &=7\times\cfrac{6}{7}\\ &=6 \end{aligned}$