Let $\langle S \rangle_R$ be the ideal of $R$ generated by the set $S$.
$1.R=\mathbb{Z},S=\{105,70,42,30\}$
$2.R=\mathbb{Z} \times \mathbb{Z},S=\{(4,3),(6,5)\}$
$3.R=\mathbb{Z}_{101}, S=\{[75]_{\equiv 101}\}$
Now I know that $\langle S \rangle_R$ is the set of the sum of all $r_i s_i$ with $s_i \in S,r_i\in R$. Now I need to check for all three examples whether $\langle S \rangle_R=R$ is true. I really don't know where to start. Up to now I have only got the tip that in the first example all elements of $S$ are divisible by $7$ except $30$.
Any help would be appreciated!
Hint: An ideal is the whole ring if and only if it contains the multiplicative identity 1. How could we use properties of the gcd to deduce if this is true in each case?
Solution a) We know that $$\gcd(105, 70) = 35 = 7 \cdot 5$$ $$\gcd(42, 30) = 6 = 2 \cdot 3$$ And thus by the Euclidean algorithm, we can write $$35 = 105x + 70y$$ $$6 = 42u + 30v$$ for some $x, y, u, v \in \mathbb{Z}$. Since $\langle S \rangle $ is an ideal of $\mathbb{Z}$ we deduce that $6, 35 \in \langle S \rangle$. Applying the same method again, $$\gcd(6, 35) = 1$$ and so $1 \in \langle S \rangle$, implying that $ \langle S \rangle = R$.
Hint b) you will need to do a similar approach with the gcd on each of the two sides to decide if all of $\mathbb{Z}$ is spanned in each dimension.
Hint c), try and make use of the fact that $101$ is prime to assist you.