#
Problem
of the Week

Problem
C and Solution

Cycles
of Eclipses

## Problem

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 \(n\).

## Solution

Since total solar eclipses happen every \(240\) days on this planet, it follows that
\(240\) is the *least common
multiple* (LCM) of \(16\), \(12\), and \(n\).

To determine the value of \(n\), we
will rewrite each of \(16\), \(12\), and \(240\) as a product of prime numbers. This
is known as *prime factorization*. \[\begin{aligned}
16 &= 2 \times 2 \times 2 \times 2\\
12 &= 2 \times 2 \times 3\\
240 &= 2 \times 2 \times 2 \times 2 \times 3 \times 5
\end{aligned}\] The LCM is calculated by determining the greatest
number of each prime number in any of the factorizations, and then
multiplying these numbers together. From the prime factorizations of
\(16\) and \(12\), we can determine that their LCM is
equal to \(2 \times 2 \times 2 \times 2 \times
3 =48\). Since \(240\) has an
extra factor of \(5\), and \(240\) is the LCM of \(16\), \(12\), and \(n\), it follows that \(5\) must be a factor of \(n\). The only number with a factor of \(5\) that is greater than \(10\) but less than \(20\) is \(15\).

Since the prime factorization of \(15\) is \(15=3
\times 5\), we can conclude that the LCM of \(16\), \(12\), and \(15\) is \(240\), as desired. Therefore \(n=15\).

**Extension:** Research
what conditions must occur for there to be a total solar eclipse on
Earth. How often do total solar eclipses occur on Earth?