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Difference Between GCF and LCM

lcmGCF vs. LCM

The Greatest Common Factor (or the GCF) is the greatest real number shared between two integers. What makes this number a factor is that it is a whole, real number that two integers share –that is, when broken down to their lowest multiples, the largest integer that’s shared between the two numbers is their greatest common factor.

On the other hand, the Lowest Common Multiple (or LCM) is the integer shared by two numbers that can be divided by both numbers. Basically, in the list of two numbers’ respective list of multiples, the lowest number that the two numbers share is their lowest common multiple.

As regards the GCF, the greatest common factor must be a prime number –that is, a number that can only be divided by itself and 1. For example, the numbers 10 and 15 are broken down as such:

10: 1, 2, 5
15: 1, 3, 5, 15

When we take both sets of factors into consideration it is plain to see that the greatest prime integer shared by both numbers is 5 –it can only be divided by itself and 1 and it shows up in both 10 and 15.

However, as regards the LCM, the number must be composite (that is, it can be divided by at least itself, 1, and another multiple). Most likely the other multiple is shared between both numbers. For example, when creating a list of the multiples of 6 and 9:

6: 6, 12, 18, 24, 30…
9: 9, 18, 27, 36, 45…

As we can see, the lowest integer shared by both 6 and 9 is 18 –it is divisible by 1, 6, 9, and itself.

The biggest difference between the GCF and the LCM is that one is based upon what can divide evenly into two numbers (GCF), while the other depends on what number shared between two integers can be divided by the two integers (LCM). One must also consider if the numbers only share itself and 1 as common multiples of factors, than those numbers are not related to each other. That’s exactly what the GCF and LCM finds –how two whole numbers relate to each other.

Summary:
1. The GCF is based upon what integer divides evenly into two numbers; the LCM is based upon what integer two numbers share in a list of multiples.
2. The GCF must be a prime number; the LCM must be a composite number.


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3 Comments

  1. How do you compute the start-up costs for starting a website?

  2. This article has a small error. The GCF does NOT have to be a prime number! For example the GCF of 64 and 56 is 8 and 8 is a composite number! Please fix the error

  3. Thanks for this it helped a lot

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