Consistently divisible

Since $N$ is divisible by $15$ , it must also be divisible by $\frac{15}{5} = 3$ , and since $N$ is divisible by $16$ , it must also be divisible by $\frac{16}{2} = 8$ .

Therefore, since $N$ is divisible by both $3$ and $8$ , it must also be divisible by $3 \cdot 8 = \boxed{24}$ .

$\text{14×15×16×17 = 57120}$

$\text{57120 not divisible by (25)}$ , thus the answer is not $\boxed{\color{#D61F06}{(25)}}$

$\text{57120 not divisible by (26)}$ , thus the answer is not $\boxed{\color{#D61F06}{(26)}}$

$\text{57120 not divisible by (27)}$ , thus the answer is not $\boxed{\color{#D61F06}{(27)}}$

$\text{57120 divisible by (24)}$ , thus the answer is $\boxed{\color{#20A900}{(24)}}$