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Lesson 1: Integers III

October 1, 2010

I will not treat some concepts you know perfectly: factor, divisor, divisibility criteria, prime and composite number etc.

However, I provide you some links to revise them at the end of the post.

 

7. Prime factor decomposition

Prime factor decomposition of a number means writing it as a product of prime factors using powers.

This expression is unique for each number. For instance:

240 = 24 · 3 · 5

924 = 22 · 3   7 · 11 

 

8. Highest common factor (H.C.F) and lowest common multiple (L.C.M)

In this course, we will find H.C.F and L.C.M of big numbers by using their prime factor decomposition.

8.1. Highest common factor (H.C.F)

The Highest Common Factor (H.C.F) of two (or more) numbers is the largest number that divides evenly into both numbers. In other words, the H.C.F is the largest of all the common factors.

FINDING THE H.C.F. OF BIG NUMBERS

 For larger numbers you can use the following method:

  1. Find prime factor decomposition of all numbers.
  2. Find which factors are repeating in all the numbers and choose them with the lowest exponent they appear.
  3. Multiply them to get  H.C.F.

Example:

Find the Highest Common Factor (H.C.F.) of 240 and -924.

240 = 24 · 3 · 5
924 = 22 · 3   7 · 11

Common factors are 2 and 3 but we have to choose the lowest exponent, this is 22 and 3.

Multiply the factors which repeat in both numbers  to get the H.C.F.

The Highest Common Factor is :  22 · 3 = 12

We write:

H.C.F(240,-924)= H.C.F(240,924)=12

8.2. Lowest common multiple (L.C.M)

The Least or Lowest Common Multiple of several integer numbers is the smallest positive number that is a multiple of all those numbers at the same time, except for 0.

The simple method of finding the L.C.M of smaller numbers is to write down the multiples of the larger number until one of them is also a multiple of the smaller number.

FINDING L.C.M. OF BIG NUMBERS

  1. Find all the prime factors of both numbers.
  2. Find which factors there are in all the numbers  and choose them with the highest exponent they appear.
  3. Multiply them to get  L.C.M.

Example:

Find the Lowest Common Multiple (L.C.M.) of 240 and -924.

From the example of finding the H.C.F. we know the prime factors of both numbers.

240 = 24 · 3 · 5
924 = 22 · 3   7 · 11

Factors are: 2, 3, 5, 7, 11.

We have to choose them with the highest exponent: 24, 3, 5, 7, 11.            

The L.C.M (240, -924) = L.C.M (240, 924)=  24 · 3   5 · 7 · 11 = 18480

Practise these concepts as well as word problems on them on the following links

Factors

Divisibility rules

Prime or composite

Prime factorization

Greatest common factor

Least common multiple

GCF and LCM: word problems

3 comments

  1. hi monica put in youtube integer rap and there is two boys with a tshirt with a cicle red and this rap is better than the other


  2. Numéros enteros (deletreado)
    I don’t know… (Maybe: one your or yours!)
    Serás un estrella
    Números enteros (deletreado)
    Este es el tipo de números que somos
    Positivo,negativo y el cero (2 veces)
    Llega a conocernos, serás un héroe de las matemáticas
    Sumando o restando, utilizando la recta numérica y… ( chips??)
    I don’t know…
    Multiplicando y dividiendo, utiliza las reglas
    No lo eches en falta, utiliza estas reglas

    Números enteros (deletreado)
    I don’t know… (Maybe: one your or yours)
    Serás una estrella
    Números enteros (deletreado)
    Este es el tipo de números que somos
    Positivo, negativo y el cero (2 veces)
    Llega a conocernos, serás un héroe de las matemáticas.
    Multiplicando y dividiendo usa las reglas,
    no lo eches en falta, usa estas reglas


  3. E-N-T-E-R-O-S
    I don´t know.
    Serás una estrella.
    E-N-T-E-R-O-S
    Estas son las clases de números que somos: positivos, negativos, y cero. (bis)
    I don´t know.
    Legando serás un héroe de las matemáticas.
    Sumando, restando, usando la linea numérica y ¿Patatas?
    Si vas muy rápido el profesor va a flipar y flipar y flipar y flipar y flipar.
    Multiplicando y dividiendo usando las reglas.
    Estas son las clases de números que somos: positivos, negativos y ceros. (bis)
    E-N-T-E-R-O-S
    Multiplicando y dividiendo usa las reglas.



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