WEBVTT 00:00:00.850 --> 00:00:08.780 The FIFA World Cup takes place every four years, and the US Census is taken every 10 years; both took place in 2010. 00:00:09.140 --> 00:00:11.790 When is the next year in which both events will happen? 00:00:12.470 --> 00:00:17.670 Okay, so we’ve been given some information about how often two specific events take place. 00:00:17.760 --> 00:00:23.320 We have the FIFA World Cup taking place every four years and the US Census taking place every 10 years. 00:00:23.490 --> 00:00:26.690 We’re also told they both take place in 2010. 00:00:26.870 --> 00:00:30.780 There are two different ways we can calculate the next year in which both events will happen. 00:00:30.880 --> 00:00:32.990 We’ll consider both of these methods. 00:00:33.220 --> 00:00:37.420 The first method is to list out the years in which each event takes place. 00:00:37.540 --> 00:00:40.230 We know they both took place in 2010. 00:00:40.340 --> 00:00:46.100 Since the FIFA World Cup takes place every four years, we know the next event is in 2014. 00:00:46.230 --> 00:00:48.130 The next is in 2018. 00:00:48.250 --> 00:00:52.290 We add four again and we find that the next event takes place in 2022. 00:00:52.430 --> 00:01:00.470 And if we continue this process, we find that the next four events take place in 2026, 2030, 2034, and 2038. 00:01:00.640 --> 00:01:03.300 We’ll repeat this process with the US Census. 00:01:03.300 --> 00:01:05.800 But this time, that takes place every 10 years. 00:01:05.980 --> 00:01:09.740 So adding 10 years, we find the next one occurs in 2020. 00:01:09.810 --> 00:01:14.860 The next in 2030 and the one after that takes place in 2040. 00:01:14.990 --> 00:01:18.700 We’re looking to find the next year in which both events will happen. 00:01:18.990 --> 00:01:22.700 In other words, the next year that appears on both lists. 00:01:22.790 --> 00:01:26.780 We can see the next year that appears in both of our lists is 2030. 00:01:26.900 --> 00:01:31.320 So the next year in which both events will take place is indeed 2030. 00:01:31.710 --> 00:01:33.650 But what about the alternative method? 00:01:33.780 --> 00:01:36.790 Well, this involves finding the least common multiple. 00:01:36.970 --> 00:01:46.460 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. 00:01:46.690 --> 00:01:54.080 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. 00:01:54.260 --> 00:01:58.330 Or alternatively, we can represent each number as a product of its prime factors. 00:01:58.430 --> 00:02:00.910 Four can be written as two multiplied by two. 00:02:01.110 --> 00:02:02.450 They are both prime numbers. 00:02:02.520 --> 00:02:05.030 And 10 can be written as five multiplied by two. 00:02:05.060 --> 00:02:06.980 They are also both prime numbers. 00:02:07.130 --> 00:02:10.280 We can then sort these prime numbers into a Venn diagram. 00:02:10.440 --> 00:02:13.510 We see that each list has one two in it. 00:02:13.690 --> 00:02:17.380 Four has a second two, and 10 has the number five. 00:02:17.490 --> 00:02:24.230 The greatest common factor of four and 10 is then the product of any prime factors in the overlap or the intersection. 00:02:24.430 --> 00:02:27.570 So we find that the greatest common factor of four and 10 is two. 00:02:27.690 --> 00:02:28.950 That’s not what we want though. 00:02:29.050 --> 00:02:31.080 We want the lowest common multiple. 00:02:31.250 --> 00:02:33.990 That’s the product of all numbers in our Venn diagram. 00:02:34.100 --> 00:02:37.070 It’s two times two times five, which is 20. 00:02:37.200 --> 00:02:40.890 So the least common multiple of four and 10 is 20. 00:02:41.050 --> 00:02:46.050 And this means both events take place in the same year every 20 years. 00:02:46.190 --> 00:02:49.370 We add 20 then to 2010. 00:02:49.480 --> 00:02:54.120 And once again, we find that both events take place in the year 2030.