Unknown Numbers and a Venn Diagram

Date: 11/26/2001 at 16:21:23
From: Christina
Subject: How to find an unknown numbers using the Venn diagram method

The GCF of two numbers is 20 and the LCM is 840. One of the numbers 
is 120. Explain how to find the missing number. (You must use the 
Venn diagram method to illustrate.)

I tried a guess-check method. I ramdomly picked some numbers (200, 
180) and I still can't find the numbers.

Date: 11/26/2001 at 17:14:40
From: Doctor Peterson
Subject: Re: How to find an unknown numbers using the Venn diagram 
method

Hi, Christina.

I'm not sure how you are expected to use a Venn diagram; my first 
thought would be to use it as in a logic problem, but that would 
require somehow showing all numbers whose GCF with 120 is 20 in one 
circle, and all numbers whose LCM with 120 is 840 in the other. That 
doesn't seem to help solve the problem.

Have you been shown a method for finding GCF and LCM that itself 
involves a sort of Venn diagram? I can imagine one, though it requires 
twisting the normal use of sets a bit. Suppose we want to find the GCF 
and LCM of 120 and 100. We can write the prime factorization of 120 in 
one circle and the factorization of 100 in another, treating repeated 
factors as separate elements, perhaps by imagining them colored 
differently:

    A = {2 2 2 3 5}

    B = {2 2 5 5}

The intersection of these two sets is {2 2 5}:

     ___A___
    /       \
    2 3 2 2 5 5
        \_____/
           B

The product of the numbers in set A is the first number, 120; the 
product of the numbers in set B is 100; the product of the numbers in 
the intersection, 2*2*5, is the GCF of the two numbers, since it 
contains every factor that is common to both (with only two 2's 
because 100 has only two factors of 2 in its prime factorization). The 
LCM is the product of factors in the union of the sets. Do you see why 
this is true?

Now do you see how you can find a set B such that the LCM is 840?

     ___A___
    /       \
    2 3 2 2 5 ? ?
        \_______/
            B

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