Relatively prime to $42$ and $70$

Question

How many numbers are relatively prime to $42$ and $70$?

There's no set limit (i.e. numbers relatively prime must be less than $42$ or $70$), so I'm unsure how to figure this out. I think I'm overthinking this problem! Any hints?

(This came from a primitive roots of unity problem.)

Answer 1

Not sure if this is what you are aiming at, but looking at the prime factorizations of $42$ and $70$ a number $n$ is relatively prime to both numbers if and only if gcd$(n,2 \cdot 3 \cdot 5 \cdot 7)=1$. Now take $n$ to be any power of $11$ for example, or any prime larger than $7$. This gives you an infinite set.

Answer 2

Considering your original problem as mentioned in comments:

Primitive $42$th roots of unity are the the set $\{e^{\frac{2\pi\iota k_1}{42}\ }\ |\ (k_1,42)=1\}$ and set of primitive $70$th roots of unity is $\{e^{\frac{2\pi\iota k_2}{70}\ }\ |\ (k_2,70)=1\}$.

Now when $e^{\frac{2\pi\iota k_1}{42}}=e^{\frac{2\pi\iota k_2}{70}} where (k_1,42)=1$ and $(k_2,70)=1$

Zero.