The number system is one of the most basic and integral parts of mathematics from basic to advanced level. In mathematical operations, Greatest Common Factor (GCF) and Lowest Common Multiple (LCM) are the most useful to simplify a fraction. These mathematical methodologies help us to find the solutions of fractions, ratios, and numerous operations. Be it adding or simplifying fractions, a basic knowledge of GFC and LCM is all we need.

**GCF vs LCM**

The basic difference between GCF and LCM is that GCF finds the greatest factor which is common to a given set of numbers. The factor means a number that divides other numbers and leaves zero (0) as its remainder. And as for LCM, it is the lowest multiple, common to a set of numbers. Multiple is something that is divided by another number with no remainder.

GCF is a widely used maths technique that is mostly learned in elementary school and continued to be used constantly after then. GFC helps to reduce a set of bigger numbers into a form that is smaller and simpler. Also during the process of factorization in the case of algebraic expressions, a GFC is found which is used to simplify the question.

LCM is yet another most important technique developed by mathematicians. It is also learned at the elementary level as soon as the teaching of fractions starts. LCM is used to add or subtract fractions that do not have a common denominator (these kinds of fractions are also called unlike fractions). An LCM is taken off the denominators concerned and the fractions are thus added.

**Comparison Table Between GCF and LCM**

Parameter of Comparison | GCF | LCM |

Uses in mathematics | These are used for simplification. | These are used to add fractions that are unlike. |

Procedures dealt with | It deals with factors, which are numbers that divide a bigger number with no reminder. | These deal with multiples, which are bigger numbers and can be divided by smaller numbers without any remainder. |

Result type | It gives a smaller result than lcm. | It gives larger results than GCF. |

How numbers are taken | While finding GCF, the numbers are taken separately. | While funding LCM, the numbers are taken together. |

What it includes | It only includes factors common to a given set. | It considers all the different factors while calculating results |

**What is GCF?**

GCF whose full form is Greatest common factor is one of the most widely used methods in the field of mathematics. Students learn it at an early age and apply it to solve their mathematical problems. Problems dealing with Simplification include breaking down a bigger number into its simplest and smallest form.

Problems relating to algebra include. simplification of an equation by putting the GFC outside the bracket. And finally, it can be used to solve various word problems too. GFC as its name suggests deals in factors. Factors are the numbers that can divide a bigger number into smaller parts with zero (0) as a reminder.

For example, two (2) is 6 because two divided by six don’t leave any remainder. GFC results are much smaller than the results of LCM as it finds factors. For example, we can take the numbers six (6) and eight (8). If we find the list of factors of these two numbers, the factors of six(6) are two(2) and three(3), i.e. 2×3. And the factors of 8 are two(2), two(2), and two(2) i.e. 2×2×2. So, the factors which appear as common in both six(6) and eight (8) are two (2). Therefore GCF of the numbers 6 and 8 equals 2.

While finding the GCF which is also known as HCF (Highest Common Factor) we take the numbers concerned separately to make the calculation easier, rather than taking it altogether. Prime numbers (numbers that have 1 or themselves as a factor) are used as factors.

**What is LCM?**

LCM whose full form is Lowest Common Multiple is another widely used mathematical device invented to help us add fractions that do not have a common denominator (unlike fractions). It is also taught at the elementary level along with GFC as soon as the concepts of fractions kick into the course. They are also used to find out when certain events happening on the loop will coincide. And this helps in solving many word problems.

Euclid who developed or rather found out these two concepts of LCM and GCF wanted to make the study of mathematics easier. LCM as the name shows deals in Multiples. Multiples are numbers which when divided by smaller numbers, have no remainder left.

For example, we can take the numbers six (6) and eight (8). If we find the list of factors of these two numbers- The factors of six(6) are two(2) and three(3), i.e. 2×3. And the factors of 8 are two(2), two(2), and two(2) i.e. 2×2×2. So, the Lowest Common Multiple of these two numbers is 2×2×2×3 which equals 48. So, The number to which 6 and 8 can divide leaving no remainder is 48.

We can find the Lowest Common Multiple of a set of numbers together and use prime numbers (numbers with no factors except itself and one) to find the Lowest Common Multiple.

**Main** **Differences Between GCF and LCM**

- GCF is used in the simplification of a bigger number into its smaller form for easier calculation.Whereas LCM is used to add fractions with different denominator(unlike fractions).
- GCF deals in factors which are numbers that divide other bigger numbers and leave nothing as remainder. However, LCM deals with multiples,which are numbers that are divided by smaller numbers with no remainder.
- GCF results are smaller than the results of LCM as it considers factors. LCM results are greater than GFC as if considering multiples.
- For ease in finding GCF when the numbers are taken separately. But LCM can be found more easily if a table containing all numbers is taken at once.
- While calculating the results, factors that are only common to every number in the set are taken in the case of GCF. Whereas while calculating LCM, every factor which appears is taken.

**Conclusion**

Mathematics as a subject provides us with various techniques to make it easier for us to solve a particular mathematical calculation. GCF and LCM being two of the most important tools which were developed long ago are still fully functioning and very useful even in the present day. Students mostly get confused while reading these two terms, but the difference lies in their names themselves.

Learning the proper use of GCF and LCM helps us understand the basic concepts. And thus with eternal importance that’s attached to these two terms, we can solve, simplify and add fractions, equations, etc. Before indulging in the concept of factorization, tutors help us to understand these terms. In some special type problems, both may look similar. We often get confused about which to use and when to use. No doubt, this is the basis of many complex problems you may face ahead.

**Reference**

- https://pubs.nctm.org/view/journals/at/34/7/article-p17.xml
- https://www.research.ed.ac.uk/en/publications/the-effects-of-a-problem-based-learning-intervention-on-primary-s

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