Finding Least Common Multiple for Day ProblemDate: 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. |
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