Difference Between GCF and LCM

GCF refers to the greatest number that can be a factor of two whole numbers. Whereas, a lower common multiple stands for the lowest number that is a multiple of two whole numbers.

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The smallest integer that appears as a common multiple and can be divided by the given numbers is LCM. The highest integer shared by all given numbers is GCF.

GCF vs LCM

The difference between GCF and LCM is the definition and meaning of the two terms. While GCF is the short-form of Greatest Common Factor, LCM is the short-form for Lowest or Least Common Multiple.

GCF vs LCM

This seminal difference between the two concepts also leads to the emergence of other dissimilarities.

Comparison Table

Parameters of ComparisonGCFLCM
DefinitionThe largest integer that divides a group of numbers into two or more equal parts.LCM is the number that connotes the lowest common multiple of a given set of numbers.
Full-FormGreatest Common FactorLeast or Lowest Common Multiple
MeaningThe largest integer that is shared by all given numbers.The smallest integer that can be divided by the given numbers.
Number TypeAlways a prime number.Always a composite number.
Other Terms UsedGreatest Common Divider, Highest Common Factor.No other terms are used.

What is GCF?

Greatest Common Factor or GCF is the greatest whole number that divides a given number into two or more equal halves. Factors are numerical digits which when multiplied result in the given number. A common factor is a factor that is mutually shared by a given set of numbers.

When we work out the common factors of two sets of numbers, we can easily ascertain the greatest common factor for both these numbers from the list of common factors so produced. The largest common factor is the GFC for the given group of numbers.

The process of finding out GCF is quite easy. We need to begin by distilling the factors of the given set of numbers. Then we locate the common factors between them. The greatest among these common factors is the GCF.

GFC is used to simplify fractions and easy the process of calculation. GFC is also known by a couple of more names including HCF or Highest Common Factor and GCD or Greatest Common Divider.

The Listing Method can be used as follows:

Numbers given: 4 and 16,

Factors: 4 (1, 2, 4).   16 (1, 4, 16).

Common Factors: 1, 4.

GCF: 4.

The Prime Factor Method of calculation:

The prime factor method can also be used to calculate GFC easily.

Numbers given: 24 and 108

Prime Factors: 24 (2 x 2 x 2 x 3) and 108 (2 x 2 x 3 x 3 x 3)

GCF: 2 x 2 x 3= 12.

gcf

What is LCM?

LCM is the shorthand used to refer to the mathematical procedure of finding out the Least or Lowest Common Multiple for a given set of numbers. The smallest common multiple that is divisible by the set of given numbers is called LCM. The number has to be a multiple of all the given numbers.

A multiple is a number derived when one number is multiplied by another number. The process of finding out LCM can be varied. The three most known and used methods include the Prime Factorization Method, Listing Of Multiples and the Division Method.

The Prime Factorization Method includes the finding out of the prime factors of each of the given numbers. This is followed by the pairing of their common prime numbers to find the LCM.

For example:

10 and 35 are the given numbers. The prime factors are: 10 (2 x 5) and 35 (5 x 7). Thus the LCM is 2 x 5 x 7 = 70. In the Listing Method, we can begin by finding out the multiples for each given number. The least of these multiples is the LCM.

For example, numbers 11 and 44 are given. Their multiples are: 11 (11, 22, 33, 44, 55….) and 44 (44, 88, 132….). The least common multiple is 44. The third method of calculating LCM is the Division Method where each of the numbers given is divided by the smallest prime digits.

The division continues until the result produced is 1. The prime factors thus distilled through this method are multiplied to find out the LCM.

lcm

Main Differences Between GCF and LCM

  1. The main difference between GCF and LCM is in terms of their definitions. While GCF is the calculation of the highest integer that divides a number into two or more equal parts, LCM refers to the common multiple of the lowest value shared by the given numbers.
  2. The full-form of each is also different. While GCF stands for Greatest Common Factor, LCM is the abbreviation used for the Least Common Multiple.
  3. The LCM number has to be a composite number when calculated correctly. The GCF answer calculated must always be a prime number.
  4. While GCF is the largest integer shared by all the given numbers, LCM is the smallest integer that can be divided by the given numbers.
  5. GCF is known by a variety of synonymous terms like GCD or HCF, while LCM is more or less the only term used to connote the least multiple that’s common to a given set of numbers.

References

  1. https://pubs.nctm.org/view/journals/at/31/8/article-p43.xml
  2. https://pubs.nctm.org/abstract/journals/at/26/4/article-p53.xml
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