Reducto reducto reducto

For $k$ from $1$ to $301$ , we must have $\gcd (k, N+k + 20 ) = 1$ . This is equivalent to $\gcd(k, N + 20) = 1$ .

Clearly, the values of $N$ from $1$ to $281$ will not work. Neither will any of the numbers $N= 282$ to $286$ , since $n + 20$ will be composite, hence have a smaller factor. However, for $N= 287$ , we have $N+20 = 307.$ It is a prime, so $\gcd(k, N+20) = 1$ for all $k<307.$ Thus, the smallest positive integer is 287.