Introductory questions on the integers of the form $ax+by$.

Question

In the book by Harriet griffin, titled Elementary theory of numbers, there are questions in section 2.1, for which my attempt is stated below. Please vet and help :

1. Describe the set of integers 3x+6y.
   All multiples of $3$, i.e. of the $\gcd$ .

2. Use the form $ax+by$ to define a set of integers all of which are even.
   As sum of two odd or two even numbers is even, so approach should have both even or both odd numbers. It is not a must that $\gcd(a,b)=2$, as $x,y$ can be chosen suitably for getting an even number. But, to always get an even, need have even $a,b$.

3. Use the form $ax+by$ to define a set of integers that are multiplies of 5. Are all multiples of $5$ included?
   $ax+by=5$. In terms of $a,b$ all multiples should be included, as the $\gcd=5$.

4. Use the linear form in two variables to determine a set of even integers that are multiples of $5$.
   $ax+by =10$, alternatively $\forall a,b \in 2n, n \in \mathbb{Z}, ax+by =5$

5. When will the set ax+by include all the integers?
   When $\gcd=1$. Then $ax+by =1$, will include all integers. Any multiple of the same, i.e. $\forall n \in \mathbb{Z}, nax + nby = n$ will have only multiples of $n$, as $a,b$.

Answer 1

1. You are right.
2. You don't answer the question. An answer is $\{2x+2y\,|\,x,y\in\mathbb Z\}$.
3. You don't answer the question. An answer is $\{5x+5y\,|\,x,y\in\mathbb Z\}$.
4. You don't answer the question. An answer is $\{10x+10y\,|\,x,y\in\mathbb Z\}$.
5. You are right: it's when $\gcd(a,b)=1$.