Laps and Least Common Multiples

Date: 05/28/2002 at 13:02:03
From: Pushkal
Subject: Running around a track

Hi Doctor,

A, B, and C start at the same time, and in the same direction, to run 
around a circular course. A makes it in 252 seconds, B in 308 seconds,
C in 198 seconds. 

At what time will three of them reach the starting point and how many 
courses would each have completed by that time?

Pushkal

Date: 05/28/2002 at 13:29:07
From: Doctor Ian
Subject: Re: Running around a track

Hi Pushkal,

What makes this problem seem hard is the big, ugly numbers.  But 
suppose we make it a smaller, tidier problem:

  A, B, and C start at the same time and in the same 
  direction, to run around a circular course.  A makes
  one circuit in 2 minutes, B in 3 minutes, and C in 5 
  minutes, and so on. 

When you look at it this way, it becomes clear that you need to 
find the smallest number of minutes that is evenly divisible by 
2, 3, and 5.  Does that make sense?  Do you see how to apply the 
same reasoning to the original problem? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/28/2002 at 13:54:30
From: Pushkal
Subject: (Continued)

Dear Sir,

Why should I find the L.C.M of the numbers?

Date: 05/28/2002 at 14:11:36
From: Doctor Ian
Subject: Re: (Continued)

Hi Pushkal,

If A can run one circuit in 2 minutes, he can run 1/2 circuit in 
1 minute, right?  Similarly, B can run 1/3 circuit in 1 minute, 
and C can run 1/5 circuit in 1 minute.  So let's see where they 
are minute by minute:

  Minutes    A      B      C
  -------  -----  -----  -----
        1   1/2    1/3     1/5
        2   2/2*   2/3     2/5
        3   3/2    3/3*    3/5
        4   4/2*   4/3     4/5
        5   5/2    5/3     5/5*

and so on.  The *'s mark completed laps, i.e., minutes after 
which the number of numerator is a multiple of the denominator. 

You want to get three *'s in the same row, which is to say, you 
want to find a numerator that is divisible by all three 
denominators simultaneously.  Moreover, you want to find the 
_smallest_ such numerator. 

What would you call that number?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/28/2002 at 15:41:52
From: Pushkal
Subject: Thank you

Thank you Doctor Math. I really love your true dedication 
and way of explanation.

Pushkal