How to test if arithmetic progression is subset of another?

Question

I have two arithmetic progressions $R = \{ a + rn, n \ge 0 \}$ and $S = \{ b + sn, n \ge 0 \}$. What are necessary and sufficient conditions to test whether R is a subset of S?

Answer 1

You need $a\in S$ and $s$ divides $r$. This ist sufficient. To show that it is necessary, see that when $a=b+sn_0\in S$ and $a+r=b+sn_1\in S$ you have $r=s(n_1-n_0)$ which means $s$ divides $r$.

Answer 2

The difference of R should be less than S and it should be one of the factor of difference of S ie $rx<sx$ where $x$ is any constant. Also $a,b$ should be mutiples such that $a<b$

Answer 3

We must have: $\forall n \in \mathbb{N}: \exists m \in \mathbb{N}: b+sn=a+rm$.

Hence $m = \frac{b-a+sn}{r}$.

For $n=1$ it means that $r|(b-a+s)$ For $n=2$ : $r|(b-a+2s) $ $r|((b-a+s)+s) $ so $r|s$ $r|(b-a+s)$ and $r|s$ $\implies r|(b-a)$.

Hence $r|s$ and $r|(b-a)$ are necessary conditions.

Are there sufficient? Let $s = kr$ and $b = a+lr$ for some $k,l \in \mathbb{N}$.

For every $n \in \mathbb{N}$ we have: $b+sn = a+lr+krn = a+r(l+kn)$ , hence it is also suffictient condition.