# The LCM (3 and 243) = ? Calculate the Least (the Lowest) Common Multiple, LCM, by Using Two Methods: 1) Numbers' Divisibility and 2) The Prime Factorization

## The least common multiple

lcm (3; 243) = ?

### Method 1. The divisibility of numbers:

#### A number 'a' is divisible by a number 'b' if there is no remainder when 'a' is divided by 'b'.

#### Divide the larger number by the smaller one.

#### When we divide our numbers, there is no remainder:

#### 243 ÷ 3 = 81 + 0

#### ⇒ 243 = 3 × 81

#### ⇒ 243 is divisible by 3.

#### ⇒ 243 is a multiple of 3.

#### The smallest multiple of 243 is the number itself: 243.

## The least common multiple:

lcm (3; 243) = 243 = 3^{5}

243 is a multiple of 3

Scroll down for the 2nd method...

### Method 2. The prime factorization:

#### The prime factorization of a number: finding the prime numbers that multiply together to make that number.

#### 3 is a prime number, it cannot be broken down into other prime factors.

#### 243 = 3^{5}

243 is not a prime number but a composite one.

** Prime number: a natural number that is only divisible by 1 and itself. A prime number has exactly two factors: 1 and itself. *

* Composite number: a natural number that has at least one other factor than 1 and itself.

### Calculate the least common multiple, lcm:

#### Multiply all the prime factors of the two numbers. If there are common prime factors then only the ones with the largest exponents are taken (the largest powers).

## The least common multiple:

lcm (3; 243) = 3^{5} = 243

243 contains all the prime factors of the number 3

### Why is it useful to calculate the least common multiple?

#### In order to add, subtract or sort fractions with different denominators, we must make their denominators the same. An easy way is to calculate the least common multiple of all the denominators (the least common denominator).

#### By definition, the least common multiple of two numbers is the smallest natural number that is: (1) greater than 0 and (2) a multiple of both numbers.

## Other similar operations with the least common multiple:

## Calculator: calculate the least common multiple, lcm

### Calculate the least common multiple of the numbers, LCM:

#### Method 1: Run the prime factorization of the numbers - then multiply all the prime factors of the numbers, taken by the largest exponents.

#### Method 2: The Euclidean algorithm:

lcm (a; b) = ^{(a × b)} / _{gcf (a; b) }

#### Method 3: The divisibility of the numbers.