## lcm (89,640; 448,200) = ?

### Approach 1. Integer numbers divisibility:

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

#### Notice that dividing our numbers leaves no remainder:

#### 448,200 ÷ 89,640 = 5 + 0;

#### => 448,200 = 89,640 × 5;

#### So, 448,200 is divisible by 89,640.

#### 448,200 is a multiple of 89,640.

#### The smallest multiple of 448,200 is the number itself: 448,200.

#### Consequently, least common multiple:

lcm (89,640; 448,200) = 448,200;

## lcm (89,640; 448,200) = 448,200 = 2^{3} × 3^{3} × 5^{2} × 83;

448,200 is a multiple of 89,640

### Approach 2. Integer numbers prime factorization:

#### Prime Factorization of a number: finding the prime numbers that multiply together to make that number.

#### 89,640 = 2^{3} × 3^{3} × 5 × 83;

89,640 is not a prime, is a composite number;

#### 448,200 = 2^{3} × 3^{3} × 5^{2} × 83;

448,200 is not a prime, is a composite number;

** Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself. *

* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.

### Calculate the least common multiple, lcm:

#### Multiply all the prime factors, by the largest exponents (if any).

#### lcm (89,640; 448,200) = 2^{3} × 3^{3} × 5^{2} × 83;

## lcm (89,640; 448,200) = 2^{3} × 3^{3} × 5^{2} × 83 = 448,200

448,200 has all the prime factors of the number 89,640

## Final answer:

Least common multiple

lcm (89,640; 448,200) = 448,200 = 2^{3} × 3^{3} × 5^{2} × 83

448,200 is divisible by 89,640. 448,200 is a multiple of 89,640.

448,200 has all the prime factors of the number 89,640

### Why do we need the least common multiple?

#### In order to add, subtract or compare fractions you have to first build their denominators the same. This common denominator is nothing else than the least common multiple of fractions' denominators, also called the least common denominator, lcd.

#### By definition, the least common multiple of two integers, LCM, is the smallest positive integer larger than 0 that is a multiple of both.

## More operations of this kind:

## Online calculator: LCM, the least common multiple

## Tutoring: what is it and how to calculate the least common multiple LCM of integer numbers

60 is a common multiple of the numbers 6 and 15, because 60 is a multiple of 6 and is also a multiple of 15. But there is also an infinite number of common multiples of 6 and 15.

#### If "v" is a multiple of "a" and "b", then all the multiples of "v" are also multiples of "a" and "b".

Common multiples of 6 and 15 are: 30, 60, 90, 120... Among them, 30 is the lowest and we say that 30 is the **least common multiple, or the lowest common multiple, or the smallest common multiple** of 6 and 15, abbreviated as **LCM**.

#### If e = LCM (a; b), then "e" contains all the prime factors involved in the prime factorizations of both "a" and "b", by the highest powers (exponents).

Based on this rule we can calculate the least common multiple, LCM, of the three numbers in the example below:

- 40 = 2
^{3} × 5 - 36 = 2
^{2} × 3^{2} - 126 = 2 × 3
^{2} × 7 - LCM (40; 36; 126) = 2
^{3} × 3^{2} × 5 × 7 = 2,520