A planet in a distant solar system has a moon and a sun. On this planet, there is a total solar eclipse whenever the following is true.
There is a full moon,
the moon is at its closest point to the planet, and
the centre of the moon is in line with the centres of the planet and the sun.
On this planet, there is a full moon every \(16\) days. Also, every \(12\) days, the moon is at its closest point to the planet. As well, every \(n\) days the centre of the moon is in line with the centres of the planet and the sun.
If \(n\) is greater than \(10\) but less than \(20\), and total solar eclipses happen on this planet every \(240\) days, determine the value of \(n\).
Theme: Number Sense