# What is the difference between LCM and HCF

If you are an A-level student studying Maths, or perhaps tackling the challenges of GCSE Maths, then you might have come across the acronyms LCM and HCF. In GCSE Maths, understanding these concepts is crucial. LCM stands for Least Common Multiple, while HCF represents the Highest Common Factor. Both concepts are essential in solving mathematical problems that involve multiple integers, particularly when it comes to division,

However, the two are often confused with each other, and it can be challenging to distinguish between them. In this blog post, we will explain the difference between LCM and HCF, their real-world use cases, and tips for understanding and remembering the differences, particularly focusing on when to use LCM or HCF in various scenarios.

**Explanation of LCM: Definition and Calculation**

The Least Common Multiple is identified as the smallest whole number found in the multiplication tables of two or more entities. It represents the smallest integer divisible by both without a remainder, a crucial concept in understanding division.

For example, to determine this value for 8 and 12, one would list the multiples obtained by multiplying 8 and 12 by other integers:

Multiples of 8: 8, 16, 24, 32, 40, 48…

Multiples of 12: 12, 24, 36, 48, 60, 72…

The smallest number in both lists is 24, so the answer is 24.

**Explanation of HCF: Definition and Calculation**

The Highest Common Factor, also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder.

For example, to find the Highest Common Factor of 8 and 12:

List the factors of each number. Factors are numbers that you can multiply together to get the original number.

- Factors of 8: 1, 2, 4, 8 (These numbers can be multiplied to get 8)
- Factors of 12: 1, 2, 3, 4, 6, 12 (These numbers can be multiplied to get 12)

Identify the shared factors, meaning numbers that are factors of both 8 and 12.

- Shared factors of 8 and 12: 1, 2, 4.
- The highest common factor of 8 and 12 will be 4.

**Key Differences between LCM and HCF**

Aspect | Least Common Multiple | Highest Common Factor |

Calculation Method | Typically calculated by listing multiples and identifying the smallest shared or through prime factorisation, taking the highest power of each prime used. | Determined by breaking down each integer into prime factors and finding the product of the highest common primes in their lowest powers. |

Nature of Results | Always equal to or larger than the largest number in the set. | Always equal to or smaller than the smallest number in the set. |

Multiples and Factors | Deals with multiples. | Focuses on factors that can divide the given integers |

**Use Cases: Where LCM is Applicable**

The LCM has broad applications in Maths, especially in fractions, percentages, and ratios. In fractions, we need to find the LCM of the denominators before we can add or subtract the fractions.

**Use Cases: Where HCF is Applicable**

The HCF is useful for tasks such as simplifying fractions and algebraic expressions, calculating the highest common factor, often involving a method that uses division.

**Real-World Scenarios: When to Apply LCM or HCF**

LCM can be used to find the duration of movies or songs that have different start and end times. By doing so, we can determine the length of the entire movie or song.

While used in computer algorithms, where it is essential to find the highest common factor of two given numbers, HCF plays a fundamental role. This concept is particularly crucial in cryptographic algorithms, where it aids in calculating private and public keys.

**Misconceptions about LCM and HCF**

One misconception is that these mathematical concepts only apply to whole numbers. However, they can also be applied to fractions, decimals, and algebraic expressions.

Another misconception is that they cannot be the same value, which is not true, particularly when dealing with denominators.

**Tips for Understanding and Remembering the Differences**

To remember the differences between LCM and HCF, you can use the acronym "SAM," which stands for "Smallest" and "Multiple" for LCM and "HAS” which stands for "Highest" and "Shared" for HCF.

Also, LCM deals with multiples, while HCF focuses on factors that divide the numbers. Understanding this distinction is a crucial step in grasping these concepts, especially when applying the method of division to problems.