elementary number theory - Prove that among any five consecutive positive integers there is one integer which is relatively prime to the other four integers.

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.