Answer
$N = 69$ (or $70$)
Work Step by Step
The transverse field in a solenoid of radius $a$ is given by :
$B = \frac{\mu}{2} \frac{NI a^2}{(a^2+x^2)^{\frac{3}{2}}}$
where $x$ is the distance from the center of the solenoid.
Then, use $\mu = \mu_0$, $I = 2.5 A$, $x = 0.06m$, $B = 6.39\times 10^{-4} T$, $a = 0.06 m$, you find $N = 69.036$
You can either pick $N = 70$ to have a magnetic field a little bit larger. Or you can pick $N = 69$ because the obtained magnetic field is closer to the one that is given.
