Question 63
For a positive integer $N$, let $c_{1} \lt c_{2} \lt \dots \lt c_{m}$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \geq 3$ such that
$$
\gcd (N, c_{i} + c_{i + 1}) \neq 1
$$
for all $1 \leq i \leq m - 1$.
Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.
$$
\gcd (N, c_{i} + c_{i + 1}) \neq 1
$$
for all $1 \leq i \leq m - 1$.
Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.