Let $a_1,\ldots,a_k$ be positive integers, such that $a_1\cdot\ldots\cdot a_k+1$ is divisble by 4.
Show that at least one of the integers $a_1+1,\ldots,a_k+1$ is divisble by 4.
I'm currently studying for an exam and I'm kind of stuck on this one. I don't know where to begin with because I haven't faced a problem like this one yet. So any help or suggestion is appreciated.
Since $4\mid a_1a_2\ldots a_k+1$, the numbers $a_1$, $a_2$, …, $a_k$ must all be odd; otherwise, $a_1a_2\ldots a_k+1$ would be odd and therefore not a multiple of $4$. And if each of them was of the form $4m+1$, their product would also be of that form and therefore $a_1a_2\ldots a_k+1$ would be of the form $4m+2$, and so would not be a multiple of $4$.
So, one of the numbers $a_j$ must be of the form $4m+3$, and therefore $4\mid a_j+1$.