A standing wave is established on a string that is clamped at one...

For a string clamped at one end and free at the other end, a node is formed at the clamped end and an antinode is formed at the free end.

The distance between a node and the adjacent antinode is $\dfrac{\lambda}{4}$.

The length of the string $L$ is given by $L = (2n - 1)\dfrac{\lambda}{4}$, where $n = 1, 2, 3, \dots$

Since $(2n - 1)$ is an integer, the length of the string must be an integral multiple of $\dfrac{\lambda}{4}$.