Some Mathematics in Park

During morning walk, I saw a bench on the right side of the path, something like below image.

As I walked further, I found another bench but on the left side of the path. Something like below image.

I found that these benches were placed at some regular intervals. There were few places on the path with benches on left side and right side both. I found this quite interesting and useful. Persons walking on the path get place to sit and relax at some intervals with views of right side or left side. Whereas, there are places where people can sit in front of each other and relax. 

I decided to get some idea about the distance between two consecutive benches on either sides of the path. I counted the number of steps required to move from one bench to other and got estimates of intervals between two consecutive benches. Left side benches were placed at interval of 200 meters whereas right side benches were 140 meters apart. 

There can be several interesting questions in this scenario. You have been sitting with your friends on benches which are in front of each other at some place on the path. You all decide to restart your walk. After what distance you would get benches facing each other again? That means, what is the minimum distance you should walk in any one direction to get benches on both sides of the path once again? Let us get it intuitively.

So, you are at a place where benches are on both sides. You start walking in one direction. After 140 meters, you get a bench on right side of the path. Here, there is no bench on the left side because that will come after 200 meters. When you reach to this bench after 200 meters from start, you don't have any bench on the right side. The next right side bench will be after 280 meters, i.e., 140+140 meters from the start. But, there, you don't have the left side bench. The next left side bench comes after 200+200 meters, i.e., 400 meters from start. But here you don't have right side bench which will be coming next after 140+140+140 = 420 meters from the start.  So, would you get benches on both sides again? If yes, after what distance from the start? 

As we see, right side benches come after every 140 meters. So, if we take the distance from a start point where we have benches on both sides then we get right side benches at distances of 140, 280, 420, 560, 700 meters etc. It is obvious that all these numbers are multiples of 140. Left side benches will come after every 200 meters or at distances that are multiples of 200. So, you have left side benches at 200 m, 400m, 600m, 800m, and so on. So the place where we have benches on both sides of the path is at such distance from the start which is multiple of both the numbers 140 and 200. Out of the multiples written so far, there is no common multiple. Let us check some more. Few more multiples of 140 are 840, 980, 1120, 1260, 1400, 1540 etc. Similarly, few more multiples of 200 are 1000, 1200, 1400, 1600, 1800 etc. We find that 1400 is multiple of 140 as well as 200. So, at a distance of 1400 meters from the start, we will have a bench on the right side. It will be the 10th bench on the right. There will also be a bench on the left side at that place. It will be 7th bench on the left. In fact, every common multiple of 140 and 200 will give us such numbers that give us the distance where we have benches on both sides of the path. We are interested in the next place to have benches on both sides. So, we need the lowest of all these common multiples. This is what least common multiple (LCM) does. 

We got common multiples of 140 and 200 quite easily. So, we got a place where benches are on both sides of the path. We can sit on them with friends and talk. It may not be that easy always. So, what could be the worst situation? 

You can try identifying such values of distances between two consecutive benches for right side and left side respectively such that you don't get benches on both sides of the path at the same place in next 5 kilometers. You have to work on lowest common multiple for this. 

You take any two numbers and multiply them. The product is multiple of both the numbers. Every multiple of the product is also multiple of the two numbers hence common multiple. So, getting common multiples  is no issue. But, do we have common multiples lesser than the product? In this case, we have. Product of 140 and 200 is 28000 which is a common multiple of both these numbers. But 1400, 2800, 4200, 5600, 7000, etc. are also common multiples. They are much lesser than the product 28000. We saw that there is no common multiple lesser than 1400, so that is the LCM of 140 and 200. The worst situation can be when we don't have any common multiple lesser than the product. That is, the product of the two numbers itself is the LCM. Are these numbers coprime? 

You can also check that the ratio of product of two numbers and their LCM is factor of both the numbers. So, it is a common factor. In fact, you will not get any common factor higher than that. So, it is highest common factor (HCF). 

Try experimenting with distance between two consecutive benches on either sides of the path to design a suitable sitting arrangement for elderly people walking on that path. Make necessary assumptions.