A person is travelling on a spaceship that moves at a speed of...

The proper time interval between consecutive birthday celebrations for the person on the spaceship is $\Delta t_0 = 1 \text{ year}$.

The speed of the spaceship is $v = \dfrac{5c}{13}$.

The time interval measured by the friend on Earth is given by the time dilation formula:

$\Delta t = \dfrac{\Delta t_0}{\sqrt{1 - \dfrac{v^2}{c^2}}}$

Substituting the given values:

$\Delta t = \dfrac{1}{\sqrt{1 - \left(\dfrac{5}{13}\right)^2}}$

$\Delta t = \dfrac{1}{\sqrt{1 - \dfrac{25}{169}}}$

$\Delta t = \dfrac{1}{\sqrt{\dfrac{144}{169}}}$

$\Delta t = \dfrac{1}{\dfrac{12}{13}} = \dfrac{13}{12} \text{ year}$