 # Problem of the Week Problem C and Solution Stargazing

## Problem

In a distant solar system, four different comets: Hypatia, Fibonacci, Lovelace, and Euclid, passed by the planet Ptolemy in $$2023$$. On Ptolemy, it is known that the Hypatia comet appears every $$3$$ years, the Fibonacci comet appears every $$6$$ years, the Lovelace comet appears every $$8$$ years, and the Euclid comet appears every $$15$$ years.

When is the next year that all four comets will pass by Ptolemy? ## Solution

Since the Hypatia comet appears every $$3$$ years, it will pass by Ptolemy in the following numbers of years: $$3,~6,~9,~12,~15,~18,~21,~24,~27,~30\ldots$$.

Since the Fibonacci comet appears every $$6$$ years, it will pass by Ptolemy in the following numbers of years: $$6,~12,~18,~24,~30,\ldots$$.

Therefore, both the Hypatia and Fibonacci comets will pass by Ptolemy in the following numbers of years: $$6,~12,~18,~24,~30,\ldots$$.

This happens because these numbers are common multiples of $$3$$ and $$6$$. If we want to determine when all four comets next pass by Ptolemy, we need to find the least common multiple (LCM) of $$3,~6,~8,$$ and $$15$$. We shall do this in two ways.

Solution 1

The first way to find the LCM is to list the positive multiples of $$3,~6,~8,$$ and $$15$$, until we find a common multiple in each list.

Number Positive Multiples
$$3$$ $$3,~6,~9,~12,~15,~18,~21,\ldots,~108,~111,~114,~117,~\mathbf{120},~123,\ldots$$
$$6$$ $$6,~12,~18,~24,~30,~36,~42,\ldots,~96,~102,~108,~114,~\mathbf{120},~126,\ldots$$
$$8$$ $$8,~16,~24,~32,~40,~48,~56,\ldots,~104,~112,~\mathbf{120},~128,\ldots$$
$$15$$ $$15,~30,~45,~60,~75,~90,~105,~\mathbf{120},~135,\ldots$$

Thus, the LCM of $$3,~6,~8,$$ and $$15$$ is $$120$$. Therefore, the next time all four planets will pass by Ptolemy is in $$120$$ years. This will be the year $$2143$$.

Solution 2

The second way to determine the LCM is to rewrite $$3,~6,~8,$$ and $$15$$ as a prime or a product of prime numbers. (This is known as prime factorization.)

• $$3 = 3$$

• $$6 = 2 \times 3$$

• $$8 = 2 \times 2 \times 2$$

• $$15 = 3 \times 5$$

The LCM is calculated by determining the greatest number of each prime number in any of the factorizations (here we will have three $$2$$s, one $$3$$, and one $$5$$), and then multiplying these numbers together. This gives $$2 \times 2 \times 2 \times 3 \times 5 = 120$$. Therefore, the next time all four planets will pass by Ptolemy is in $$120$$ years. This will be the year $$2143$$.

Note: The second method is a more efficient way to find the LCM, especially when the numbers are quite large.