Least Common Multiple Word Problem

Date: 12/10/96 at 06:47:35
From: Danny Hawkins
Subject: Least common multiples

My dad was helping me with my math and ended up confusing me more than 
I was to begin with with this problem:

The Plush Hotel received a shipment of glasses packed in full cartons 
of 40 glasses each.  Another shipment of glasses packed 24 to a carton 
went to the Maison Restaurant.  Glasses were also shipped to State 
University, but this shipment contained cartons with 25 glasses each.  
If the hotel, the restaurant and the university each received the same 
number of glasses and if none of them received more than a thousand 
glasses, how many glasses were in each shipment?  How many cartons 
were in each shipment?

My dad came up with 600 glasses; hotel = 15 cartons; restaurant = 25 
cartons; and university = 24 cartons.  But I have no idea how he got 
there and now he has left for work.  Please help.

Date: 12/11/96 at 21:03:43
From: Doctor Rob
Subject: Re: Least common multiples

The key words lie in the subject line of your query:  Least Common
Multiple.

However many glasses the Plush Hotel received, the number must have 
been a multiple of 40, since they must have received a whole number of 
boxes. Likewise, the number of glasses received by the Maison 
Restaurant had to be a multiple of 24, and the number received by the 
University had to be a multiple of 25.   

If these three places received the same number of glasses, call the 
number N, the N must be a multiple of 40, of 24, and of 25. In this 
situation, N is called a common multiple of 40, 24, and 25. You are 
asked to find the least common multiple, or the smallest number which 
is a multiple of all three of the numbers 40, 24, and 25.

One way to do this is to factor the three numbers into prime factors:
40 = 2^3 * 5, 24 = 2^3 * 3, and 25 = 5^2. Any common multiple must
have a divisor of 2^3 (from the first two), 3 (from the second), and
5^2 (from the third). Furthermore, if a number is of the form 
N = 2^3 * 3 * 5 * M for some whole number M, then it is a multiple of 
all three of the given numbers. When we multiply this out, we get 
N = 600*M. You need to verify that N is a multiple of all three of 
the given numbers. The smallest M which works is M = 1, so that the 
solution is, as your dad found, N = 600. You can try to figure out 
why M = 2 doesn't work.

You can also think of this process sequentially. Start with 
40 = 2^3 * 5. What do we have to multiply this by to make it a 
multiple of 24 = 2^3 * 3?  The answer is 3, since 40 is already a 
multiple of 2^3. This tells us that 120 = 40 * 3 = 2^3 * 3 * 5 is 
the least common multiple of 40 and 24. Now what do we have to 
multiply 120 by to make it a multiple of 25 = 5^2?  It is already a 
multiple of 5, so we just have to multiply it by an additional 5.  
This gives 5*120 = 600.

To find the number of cartons in each shipment, you need to divide 
600 (the total number of glasses shipped to each location) by the 
number of glasses per carton shipped to that particular location.  
So, for instance, the number of cartons shipped to the Plush hotel 
would be 600/40 = 15 cartons.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/