## The least common multiple:

lcm (54; 3,515) = 2 × 3^{3} × 5 × 19 × 37 = 189,810

The two numbers have no prime factors in common

189,810 = 54 × 3,515

## 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:

3,515 ÷ 54 = 65 + 5

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

54 ÷ 5 = 10 + 4

Step 3. Divide the remainder of the step 1 by the remainder of the step 2:

5 ÷ 4 = 1 + 1

Step 4. Divide the remainder of the step 2 by the remainder of the step 3:

4 ÷ 1 = 4 + 0

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

1 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 (54; 3,515) = 1

### 2. Calculate the least common multiple:

#### The least common multiple, Formula:

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

#### lcm (54; 3,515) =

^{ (54 × 3,515) }/_{ gcf, hcf, gcd (54; 3,515) } =

^{ 189,810 }/_{ 1 } =

#### 189,810

## The least common multiple:

lcm (54; 3,515) = 189,810 = 2 × 3^{3} × 5 × 19 × 37

### 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.