Finding Least Common Multiple for Day Problem

Date: 01/14/2004 at 19:52:32
From: Linda
Subject: calendar - when a 7day and 10day fall on same day

Club A meets every 7 days.  Club B meets every 10 days.  In how many 
days will the clubs meet on the same day?  I used a calendar to figure
this out, but there must be a mathematical solution I am not
getting... help!

Date: 01/15/2004 at 14:43:38
From: Doctor Ian
Subject: Re: calendar - when a 7day and 10day fall on same day

Hi Linda,

Sometimes a useful thing to do is try to look at a smaller version of
the problem you're trying to solve.  Suppose the intervals are 2 and 3
days, instead of 7 and 10 days.  Then we have

  Day:      1  2  3  4  5  6  7  8  9 10 11 12
  Club 1:      x     x     x     x     x     x
  Club 2          x        x                 x
                           ^                 ^
                           |_________________|__ both meet


So the clubs appear to both meet every 6 days.  But how could we have
figured that out without drawing the picture? 

Here's one way to think about it.  We can imagine blocks of days
corresponding to numbers of meetings.  For example, for the first club, 

   1 meeting  ->  2 days
   2 meetings ->  4 days
   3 meetings ->  6 days
   4 meetings ->  8 days
   5 meetings -> 10 days
   6 meetings -> 12 days

and for the second club, 

   1 meeting  ->  3 days
   2 meetings ->  6 days
   3 meetings ->  9 days
   4 meetings -> 12 days
   
What we'd like to know is:  When can we have two blocks of the same
size?  Well, the first block is always a multiple of 2, right?

   ___ meetings = ___ * 2

and the second block is always a multiple of 3, right?

   ___ meetings = ___ * 3

So what we're looking for is some number that is a multiple of both 2
and 3.  The smallest such number is called the 'least common multiple'
(or LCM) of those two numbers.  If the two numbers have no factors in
common, the LCM is just the product of the numbers.  For example, the
LCM of 7 and 11 is 

  7 * 11 = 77

and there is no number smaller than this that is a multiple of both
numbers.  But consider a couple of numbers like 6 and 10.  The LCM of
6 and 10 is 30 because this is the smallest number whose prime factors
can be used to construct both 6 and 10:


       _________________ 2 * 5 = 10
      |
  ---------
  |       |   
  2 * 3 * 5 = 30
  |   |
  -----
    |___________________ 2 * 3 = 6


Does this make sense?

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

Date: 01/17/2004 at 05:29:09
From: Linda
Subject: Thank you (calendar - when a 7day and 10day fall on same day)

Thank you so much for answering my email.  I liked your visual 
explanation, and I'll use it when explaining this concept to my kids.