Number theory proof - gcf and lcm

In summary, the conversation discusses the problem of proving the equality of gcd of the least common multiple of two numbers and a third number, with the least common multiple of the gcd of the two numbers and the gcd of the third number. The suggested solution is to use Venn diagrams to represent the prime factors involved in the lcm and gcd operations. However, it is not clear if the person is familiar with this method.
  • #1
roto25
11
0

Homework Statement


Prove gcd(lcm(a, b), c) = lcm(gcd(a, c), gcd(b, c))

I've tried coming up with a way to even rewrite it but I'm not really able to do it.
 
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  • #2
roto25 said:

Homework Statement


Prove gcd(lcm(a, b), c) = lcm(gcd(a, c), gcd(b, c))

I've tried coming up with a way to even rewrite it but I'm not really able to do it.

Hi roto! :smile:

Easiest is to set up a couple of Venn diagrams with the supposed prime factors in it.
An lcm is a union and a gcd is an intersection.

Do you know how to do that?
 

1. What is the definition of GCF (Greatest Common Factor) and LCM (Least Common Multiple)?

GCF is the largest number that divides evenly into two or more numbers. LCM is the smallest number that is a multiple of two or more numbers.

2. How do you find the GCF and LCM of a set of numbers?

To find the GCF, you can list out the factors of each number and then identify the greatest number that appears in all lists. To find the LCM, you can list out the multiples of each number and then identify the smallest number that appears in all lists.

3. Can the GCF and LCM of a set of numbers be the same?

Yes, it is possible for the GCF and LCM of a set of numbers to be the same. This occurs when the numbers in the set are all the same or when one number is a multiple of the other.

4. How are GCF and LCM useful in number theory?

GCF and LCM are important concepts in number theory because they help us understand the relationships between numbers and identify patterns. They are also useful in solving problems involving fractions and simplifying ratios.

5. Are there any shortcuts or strategies for finding the GCF and LCM?

Yes, there are several strategies for finding the GCF and LCM. These include using the prime factorization method, the ladder method, and the division method. It is important to practice and use the method that works best for you.

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