Two integers are called relatively prime if the greatest common divisor of $m$ and $n$ is $1$. Prove that among any five consecutive positive integers there is one integer which is relatively prime to the other four integers. (Hint: For any two positive integers $m < n$, any common divisor has to be less than or equal to $n - m$).
The question has already been asked before here, but I do not understand the steps. Is there a simpler approach? Or can anyone explain any of the answers provided in the linked question in details?
The problem is I'm not comfortable with Pigeon-hole principle or GCD (and divisibility) or modular arithmetic. Assume I do not know any of these.