I will not treat some concepts you know perfectly: **factor, divisor, divisibility criteria, prime and composite number** etc.

However, I provide you some links to revise them at the end of the post.

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### 7. Prime factor decomposition

**Prime factor decomposition of a number **means writing it as a product of prime factors using powers**.**

This expression is unique for each number. For instance:

240 = 2^{4} · 3 · 5

924 = 2^{2} · 3 7 · 11

### 8. Highest common factor (H.C.F) and lowest common multiple (L.C.M)

In this course, we will find H.C.F and L.C.M of big numbers by using their prime factor decomposition.

### 8.1. Highest common factor (H.C.F)

The **Highest Common Factor (H.C.F)** of two (or more) numbers is the largest number that divides evenly into both numbers. In other words, the H.C.F is the largest of all the common factors.

**FINDING THE H.C.F. OF BIG NUMBERS**

** **For larger numbers you can use the following method:

**Find prime factor decomposition of all numbers.**
**Find which factors are repeating in all the numbers and choose them with the lowest exponent they appear.**
- Multiply them to get H.C.F.

**Example:**

**Find the Highest Common Factor (H.C.F.) of 240 and -924.**** **

240 = 2^{4} · 3 · 5
924 = 2^{2} · 3 7 · 11

Common factors are 2 and 3 but we have to choose the lowest exponent, this is 2^{2} and 3.

Multiply the factors which repeat in both numbers to get the H.C.F.

The Highest Common Factor is : 2^{2} · 3 = 12

We write:

H.C.F(240,-924)= H.C.F(240,924)=12

### 8.2. Lowest common multiple (L.C.M)

**The Least or Lowest Common Multiple of several integer numbers** is the smallest positive number that is a multiple of all those numbers at the same time, except for 0.

The simple method of finding the L.C.M of smaller numbers is to write down the multiples of the larger number until one of them is also a multiple of the smaller number.

**FINDING L.C.M. OF BIG NUMBERS**

**Find all the prime factors of both numbers.**
**Find which factors there are in all the numbers and choose them with the highest exponent they appear.**
- Multiply them to get L.C.M.

**Example:**

Find the Lowest Common Multiple (L.C.M.) of 240 and -924.

From the example of finding the H.C.F. we know the prime factors of both numbers.

240 = 2^{4} · 3 · 5
924 = 2^{2} · 3 7 · 11

Factors are: 2, 3, 5, 7, 11.

We have to choose them with the highest exponent: 2^{4}, 3, 5, 7, 11.

The L.C.M (240, -924) = L.C.M (240, 924)= 2^{4} · 3 5 · 7 · 11 = 18480

Practise these concepts as well as word problems on them on the following links

Factors

Divisibility rules

Prime or composite

Prime factorization

Greatest common factor

Least common multiple

GCF and LCM: word problems