Video Transcript
The FIFA World Cup takes place every four years, and the US Census is taken every 10 years; both took place in 2010. When is the next year in which both events will happen?
Okay, so we’ve been given some information about how often two specific events take place. We have the FIFA World Cup taking place every four years and the US Census taking place every 10 years. We’re also told they both take place in 2010. There are two different ways we can calculate the next year in which both events will happen. We’ll consider both of these methods.
The first method is to list out the years in which each event takes place. We know they both took place in 2010. Since the FIFA World Cup takes place every four years, we know the next event is in 2014. The next is in 2018. We add four again and we find that the next event takes place in 2022. And if we continue this process, we find that the next four events take place in 2026, 2030, 2034, and 2038. We’ll repeat this process with the US Census. But this time, that takes place every 10 years. So adding 10 years, we find the next one occurs in 2020. The next in 2030 and the one after that takes place in 2040.
We’re looking to find the next year in which both events will happen. In other words, the next year that appears on both lists. We can see the next year that appears in both of our lists is 2030. So the next year in which both events will take place is indeed 2030.
But what about the alternative method? Well, this involves finding the least common multiple. If we can find the least common multiple of four and 10, that will tell us the number of years after 2010 in which both events will take place. And one way we can find the least common multiple is by listing the four times tables and 10 times tables and finding the smallest number that occurs in both. Or alternatively, we can represent each number as a product of its prime factors. Four can be written as two multiplied by two. They are both prime numbers. And 10 can be written as five multiplied by two. They are also both prime numbers.
We can then sort these prime numbers into a Venn diagram. We see that each list has one two in it. Four has a second two, and 10 has the number five. The greatest common factor of four and 10 is then the product of any prime factors in the overlap or the intersection. So we find that the greatest common factor of four and 10 is two. That’s not what we want though. We want the lowest common multiple. That’s the product of all numbers in our Venn diagram. It’s two times two times five, which is 20.
So the least common multiple of four and 10 is 20. And this means both events take place in the same year every 20 years. We add 20 then to 2010. And once again, we find that both events take place in the year 2030.
