15. Let the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ have its right focus at $F$. Draw a perpendicular from $F$ to one of the asymptotes of the hyperbola $C$, intersecting the hyperbola at point $M$ and the asymptote at point $N$. If $|F N|=a$, and $\overrightarrow{F M}=\lambda \overrightarrow{M N}$, then $\lambda=$
15.1.

Since $|F N|=a=b$, therefore, $c=\sqrt{2} a$.
Let $l_{F N}: y=-x+c$.
Then, solving it together with the hyperbola equation yields $M\left(\frac{3 c}{4}, \frac{c}{4}\right)$. And $N\left(\frac{c}{2}, \frac{c}{2}\right)$, so $\overrightarrow{F M}=\overrightarrow{M N}$.
Therefore, $\lambda=1$.