# 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?