A2. What is the smallest natural number $n$ for which the interval $\left[\frac{n+8}{2}, \frac{2 n+14}{3}\right]$ contains at least 4 natural numbers?
(A) 16
(B) 17
(C) 18
(D) 19
(E) None of the above.
A2. The length of the interval is equal to $\frac{2 n+14}{3}-\frac{n+8}{2}=\frac{(4 n+28)-(3 n+24)}{6}=\frac{n+4}{6}$. Since the interval must contain at least 4 natural numbers, its length must be at least 3 (since both endpoints can be natural numbers). Therefore, $\frac{n+4}{6} \geq 3$ or $n \geq 14$. If $n=14$, the interval is [11,14] and contains 4 natural numbers. The correct answer is therefore (E).