# The LCM (144 and 136) = ? Calculate the Least (the Lowest) Common Multiple, LCM, by Using Two Methods: 1) The Prime Factorization of the Numbers and 2) The Euclidean Algorithm

## The least common multiple

lcm (144; 136) = ?

### Method 1. The prime factorization:

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

#### 144 = 2^{4} × 3^{2}

144 is not a prime number but a composite one.

#### 136 = 2^{3} × 17

136 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 (144; 136) = 2^{4} × 3^{2} × 17 = 2,448

The two numbers have common prime factors

## Method 2. The Euclidean Algorithm:

### 1. Calculate the greatest (highest) common factor (divisor):

#### This algorithm involves the process of dividing numbers and calculating the remainders.

#### 'a' and 'b' are the two natural numbers, 'a' >= 'b'.

#### Divide 'a' by 'b' and get the remainder of the operation, 'r'.

#### If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.

#### Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.

#### Step 1. Divide the larger number by the smaller one:

144 ÷ 136 = 1 + 8

Step 2. Divide the smaller number by the above operation's remainder:

136 ÷ 8 = 17 + 0

At this step, the remainder is zero, so we stop:

8 is the number we were looking for - the last non-zero remainder.

This is the greatest (highest) common factor (divisor).

#### The greatest (highest) common factor (divisor):

gcf, hcf, gcd (144; 136) = 8

### 2. Calculate the least common multiple:

#### The least common multiple, Formula:

#### lcm (a; b) = ^{ (a × b) } / _{ gcf, hcf, gcd (a; b) }

#### lcm (144; 136) =

^{ (144 × 136) }/_{ gcf, hcf, gcd (144; 136) } =

^{ 19,584 }/_{ 8 } =

#### 2,448

## The least common multiple:

lcm (144; 136) = 2,448 = 2^{4} × 3^{2} × 17

### 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.