# Math 1: Least Common Multiple and Greatest Common Divisor

*A Biomedical science student, certificated in english, spanish, french and japanese language, who also writes diverse topic articles.*

Content• What is the Least common multiple.

• What is the Greatest common divisor.

• What are the prime numbers.

• Practice exercises.

• Answers.

• Extra material.

## What is the Least common multiple

The least common multiple is that value that, as its name indicates, is the smallest multiple of two or more numbers at the same time.

To obtain the LCM, prime factorization is used, that is, when a number is taken and divided by the largest possible prime number, doing the same with the quotient obtained until it reaches 1.

In the case of the least common multiple, after factoring by prime numbers, all the factors are taken, whether common or not, in the case of commons, the one with the greatest exponent will be taken to perform a multiplication and obtain a number integer, which is the LCM.

In the example above, the numbers 180 and 324 are factored and it is obtained that 180 = (2^{2}) (3^{2}) (5) and 324 = (2^{2}) (3^{4}) The common factors are 2 and 3, while the non-common factor is 5, in this case we would take 2^{2} and 3^{4}, which are the common factors with the highest exponent.

We need to multiply 2^{2}, 3^{4}, 5 in order to get the **LCM= (2 ^{2})(3^{4})(5)= (4)(81)(5)= 1620**.

## What is the Greatest common divisor

The greatest common divisor is that value that, as its name indicates, is the greatest divisor of two or more numbers at the same time.

To obtain the GCD, prime factorization is used, that is, when a number is taken and divided by the greatest possible prime number, doing the same with the quotient obtained until reaching 0.

In the case of the greatest common divisor, you select the elements that are common in the factorization of the numbers from which you want to get the GCD, then multiply these with each other so that a whole number is obtained.

In the example above, the numbers 180 and 324 are factored, obtaining 180 = (2^{2}) (5) (3^{2}), and 324 = (2^{2}) (3^{4}). Here the common numbers are 2^{2} and 3^{2}.

So, to obtain the GCD we need to multiply 2^{2} and 3^{2} in order to ger a number.

In this case the answer is **GCD=(2 ^{2})(3^{2})= (4)(9)= 36.**

## What are the prime numbers

Prime numbers are those numbers greater than 1 that are only divisible between themselves and 1. For example the numbers 2, 3, 5, 7, 11...

These numbers are useful for common prime factorization as shown earlier in both LCM and GCF examples.

## Excercises

- Calculate the GCD of the following numbers.

- 350, 185, 1235

- 3460. 150, 339

- 1000, 189, 750

- Calculate the LCM of the following numbers.

- 45, 33, 19

- 85 , 98, 35

- 50, 20, 80

## Answers

- Calculate the GCD of the following numbers.

- 350= (2)(5^{2})(7), 185= (5)(37), 1235= (5)(13)(19)

The only common number is 5, so the **GCD= 5**

- 3460= (2^{2})(5)(173), 150= (2)(5^{2})(3), 339= (2)(13^{2})

The only common number is 2, so the **GCD= 2**

- 1000= (2^{3})(5^{3}), 180= (2^{2})(5)(3^{2}), 750= (2)(5^{3})(3)

The only common numbers are 2 and 5, so the **GCD= (2)(5)= 10**

- Calculate the LCM of the following numbers.

- 45= (5)(3^{2}), 33= (3)(11), 19= (19)

The common number is 3, while the uncommon numbers are 5, 11 and 19. So to calculate the LCM we take both and multiply them.

**LCM= (3**^{2}**)(5)(11)(19)= (9)(5)(11)(19)= 9405**

- 85= (5)(17), 98= (2)(7^{2}), 35= (5)(7)

We have no common number in here, the uncommon numbers are 5, 17, 2 and 7. So to calculate the LCM we take both and multiply them.

**LCM= (7**^{2}**)(5)(17)(2)= (49)(5)(17)(2)= 8330**

- 50= (2)(5^{2}), 20= (2^{2})(5), 80= (2^{4})(5)

The common numbers are 2 and 5, while there's no uncommon numbers. So to calculate the LCM we take both and multiply them.

**LCM= (2**^{4}**)(5 ^{2})= (16)(25)= 400**

**© 2021 Daniela Alejandra Rodríguez Cerda**