GCF vs LCM: Difference and Comparison

The greatest number of GCF can be a factor of two whole numbers. A lower common multiple stands for the lowest number, a multiple of two whole numbers.

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

Key Takeaways

1. GCF, or greatest common factor, is the largest number that divides evenly into two or more integers; LCM, or least common multiple, is the smallest number that is a multiple of two or more integers.
2. GCF is used to simplify fractions and solve equations involving integers; LCM is used to find a common denominator for fractions and solve multiples problems.
3. GCF is calculated by finding the common factors of the given integers and choosing the largest one; LCM is calculated by finding the prime factors of the given integers and multiplying the highest power of each factor together.

GCF vs. LCM

GCF (Greatest Common Factor) is the largest number that can divide two numbers completely without any remainder. For example, the GCF of 24 and 36 is 12. LCM (Least Common Multiple) is a mathematical way of finding the smallest multiple of two or more numbers. For example, the LCM of 6 and 10 is 30.

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

Comparison Table

Parameters of Comparison	GCF	LCM
Definition	The 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-Form	Greatest Common Factor	Least or Lowest Common Multiple
Meaning	The largest integer that is shared by all given numbers.	The smallest integer that can be divided by the given numbers.
Number Type	Always a prime number.	Always a composite number.
Other Terms Used	Greatest Common Divider, Highest Common Factor.	No other terms are used.

What is GCF?

The 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 that, when multiplied, result in the given number. A given set of numbers mutually shares a common factor.

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 ease the process of calculation. Several names, including HCF, Highest Common Factor, GCD, or Greatest Common Divider, are also known as GFC.

The Listing Method can be used as follows:

Numbers are 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 are 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.

What is LCM?

LCM is the shorthand used to refer to the mathematical procedure of finding the Least or Lowest Common Multiple for a given set of numbers. The smallest common multiple 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 derived when one number is multiplied by another. The process of finding out LCM can be varied. The three most known and used methods include the Prime Factorization Method, the 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 pairing 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 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 number 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 the 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 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 various synonymous terms like GCD or HCF, while LCM is more or less the only term used to connote the least multiple 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

Last Updated : 11 June, 2023