This picture
shows 5 people playing a game in which the players have to pass the ball to the third person after them in the direction of the arrow. The game starts with Louis.
So in the first round Louis passes the ball to Thomas .
In the second round Thomas passes the ball to Michaelle.
In the 99th round who will the ball end up ?
I have read on the answers section that the answer is worked out by finding the LCM of three and five.
Which is fifteen.And that the ball will be with Louis every fifth round.
I'm not sure how they reached that conclusion . Why does the LCM matter?
Edit
So we have 3 times table :3,6,9,12,15 And 5 times table 5,10,15,15
And the LCM is 15
From this how do we deduce that the ball will be with Louise every fifth round?
Also hoe can I use modular arithmetic to solve this? What I did was 3*98=294 . The 294 =4 |mod5| But that answer is wrong. The correct answer is Mark.
The first question to ask yourself is, "How many rounds does it take for the ball to get back to Louis?" Suppose that Louis is at position $0$. If we just keep adding $3$ every time the ball is passed, it will get back to Louis for the first time when that number is divisible by $5$. This is where the LCM comes in. $\text{LCM}(3,5)=15,$ so the ball gets back to Louis after $5$ passes. So it will be back to Louis after $95$ passes, and we just need to know where it will be after $4$ more passes. If we add $3$ four times we get $12$ which is congruent to $2$ modulo $5$, and the ball is at position $2,$ that is with Mark.