What is the smallest positive integer $N$ such that $N / 3, N / 4, N / 5, N / 6$ and $N / 7$ are all integers?
(a) 420
(b) 350
(c) 210
(d) 300
(e) 280
The correct option is (a).

For $\frac{N}{3}, \frac{N}{4}, \frac{N}{5}, \frac{N}{6}$ and $\frac{N}{7}$ to be integers, $N$ must be a common multiple of $3, 4, 5, 6$ and 7. Since we want the smallest possible $N$, it must be the least common multiple (LCM) of 3, 4, 5, 6 and 7, that is,

$$
N=3 \times 4 \times 5 \times 7=420
$$