$(c,d,N)=1$ then there exist $c'=c+tN$ and $d'=d+sN$ for some integers $s,t$ such that $(c',d')=1$

Question

In the marked area where its written $(c_1,N)=1,$ I feel its wrong.What should be the modified proof?

Answer 1

Write $c=\prod_pp^{n_p}$ for some integers $n_p$. Then take $c_2=\prod_{p\mid N}p^{n_p}$, namely, the product of all prime powers in the prime factorization of $c$ that are not relatively prime to $N$. Then clearly $\gcd(c_1,N)=1$, and the rest of the proof goes through: note as well that if a prime $p$ divides $c_2$, then by definition $p$ divides $N$.

---

Hope this helps.