1992 AMC 12 Problem 23

Group the integers by their residues modulo $7$

Pair residue classes $1$ with $6,$ $2$ with $5,$ and $3$ with $4$; treat residue $0$ separately

The residue classes $1,2,\ldots,6,0$ contain $8,7,7,7,7,7,7$ numbers, respectively. We may take numbers from at most one class in each complementary pair $(1,6),(2,5),(3,4),$ and at most one multiple of $7.$ Thus $$\begin{aligned} |S|&\le\max(8,7)+\max(7,7)\\ &\qquad+\max(7,7)+1=23. \end{aligned}$$ Taking every number whose residue modulo $7$ is $1,$ $2,$ or $3,$ together with one multiple of $7,$ attains $23.$

Thus the correct answer is E.