Let $R$ be the set of primitive $42^{\text{nd}}$ roots of unity, and let $S$ be the set of primitive $70^{\text{th}}$ roots of unity. How many elements do $R$ and $S$ have in common?
GCD(42,70) = 14, so ...
@above it says primitive...
To find the number of elements that $R$ and $S$ have in common, we need to find the greatest common divisor (gcd) of $42$ and $70$.
We can find the gcd using the Euclidean algorithm. Starting with $70$ and $42$, we can divide $70$ by $42$ to get a quotient of $1$ and a remainder of $28$.
Next, we divide $42$ by $28$ to get a quotient of $1$ and a remainder of $14$.
Continuing, we divide $28$ by $14$ to get a quotient of $2$ and a remainder of $0$.
Since the remainder is $0$, the gcd of $42$ and $70$ is $14$.
Therefore, $R$ and $S$ have $\boxed{14}$ elements in common.
To find the number of elements that $R$ and $S$ have in common, we need to find the greatest common divisor (gcd) of $42$ and $70$.
The primitive $42^{\text{nd}}$ roots of unity are solutions to the equation $z^{42} = 1$, and the primitive $70^{\text{th}}$ roots of unity are solutions to the equation $z^{70} = 1$.
The number of distinct solutions to an equation of the form $z^k = 1$ is equal to the gcd of $k$ and the number of complex solutions to the equation. This is a result of Euler's totient theorem.
So, we need to find the gcd of $42$ and $70$. Let's prime factorize $42$ and $70$:
$42 = 2 \cdot 3 \cdot 7$
$70 = 2 \cdot 5 \cdot 7$
To find the gcd, we take the product of the common prime factors raised to the power of the smallest exponent. In this case, the common prime factor is $7$, so the gcd is $7^1 = 7$.
Therefore, $R$ and $S$ have $\boxed{7}$ elements in common.