Definition 3.1.1 in page 25 of this book is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3:
Now choose an integer $w_1$ such that it is from the prescribed residue class modulo $d_2/ \gcd(d_1, d_2)$, and that for any prime divisor $p$ of $q$ not dividing $d_1d_2$, we have $w_1 \not\equiv −m/d_1\pmod p$. The existence of such integers is guaranteed by the Chinese Remainder Theorem. How holds $w_1 \not\equiv −m/d_1\pmod p$ and how it comes from CRT?
PS this is an exercise in Apostol's book Ch8 and also Montgomery's book Ch9. In Apostol "quasiperiod" is named "induced modulus".
$\forall i\!:\, w\not\equiv b\pmod{\!p_i}\!\iff\! \forall i\!:\, p_i\nmid w\!-\!b\!$ $\iff\! \forall i\!:\, (p_i,w\!-\!b)\!=\!1\!\iff\! \overbrace{\color{#0a0}{(P,w\!-\!b)\!=\!1}}^{\!\!\!\large P\ =\, \prod p_i}$
When $\,w\equiv a\pmod{\!d}\,$ so $\,w=a\!+\!nd,\,$ the above becomes $\,\color{#0a0}{(P,\,a\!-\!b+ n\,d)\!=\!1}.\,$ In OP we have $(P,d)\!=\!1,\,$ so it is solvable for $\,n\,$ by the Coprime Dirichlet Theorem (which - as shown there - is indeed provable by CRT, but also has a much simpler direct proof).