Find shared terms between 2 arithmetic sequence

Question

Question is as follows:

Find formula ($a_{n}=a_1+d(n-1)$) for shared terms of $8,11,14,...$ and $2,7,12,...$ ?

My Work :

I found $a_1$ by continuing sequences and then found $a_2$ which gave me $a_2-a_1=d$ and found the answer.

Problem:

Looked up the answer and saw something I couldn't understand (final formula was same though).

Book's approach:

$d $ is $d = 3\times5$ or, in other words, is LCM of common differences ($ LCM(d_1,d_2)$). And after that found $a_1$ by extending sequence and done.

Confusion :

That should be okay when we have no constant added (like when it's $3x$ or $5x$) because its just like LCM of $3$ and $5$ but I can't understand why it works when there is $3x+5$ and $5x-3$.

Search result:

I get Shared terms in an arithmetic sequence hint, and I've done this as my second option. Yet it's not always fast enough to find those numbers because of gaps that can get bigger than few numbers.

Answer 1

Equate the common terms: $$8+3n=2+5m \Rightarrow 5m-3n=6 \Rightarrow m=3+3k,n=3+5k$$ When $k=0$ (note $m,n\ge 0$) we get: $$8+3n=8+3\cdot 3=17\\ 2+5m=2+5\cdot 3=17$$ which is the first common term. Hence: $$8+3n=8+3(3+5k)=17+15k$$ is the resulting common arithmetic series.

Answer 2

Hint: Chinese remainder theorem.

Answer 3

The first sequence is $8,11,14,17,20,\dots$ and the second sequence is $2,7,12,17,22,\dots$. So the first shared term is $17$ and after this shared terms will occur at intervals of $\text{LCM}(3,5) = 15$.

Note that the reason we know shared terms must exist here is because $\text{GCD}(3,5)=1$. This means we can be sure that there are integers $a$ and $b$ such that $3a+5 = 5b-3$.

If the greatest common denominator of the differences is not $1$ then there may be no shared terms at all. For example, $1,3,5,7,\dots$ and $2,6,10,14,\dots$ have no terms in common.