Sunday, October 20, 2013

Factors, Multiples, GCF & LCM : Factors, Prime Factorization, Factorization

Factors: Factors of a number are those which can divide the number without remainder i.e. a number is divisible by its factors.
Factors of 12:1, 2, 3, 4, 6, 12
Factors of 30: 1, 3, 5, 6, 10 15, 30

Prime Factorization: Every integer greater than 1 that is not a prime can be written as a product of primes. This is called its prime factorization.
e.g 60 = 2x2x3x5 is the prime factorization of 60. But 60 = 4x15 is not prime factorization.

Factorization: When a integer is expressed as the product of its factor, we say that the number has been factorized and the expression is called factorization. So, in 90 = 9x10, 90 has been factorized and (9x10) is the factorization.
From the given examples you may think that factorization is an easy process. But actually, it can be very tough. Given an unfamiliar number (e.g 327531913) it is very tough to find out its factorization. As the number gets bigger & bigger (300 digit) it becomes practically impossible to factorize it. This property of factorization is used in cryptography to provide security in computer systems.

GCF: Greatest Common Factor
Factor of 16: 1, 2, 4, 8, 16
Factor of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factor of 16 and24 are 1, 2, 4, 8
The greatest is 8
:. GCF (Greatest Common Factor) is 8
It can be sown by Venn Diagram also.

Simplest way to find out GCF: We will try to illustrate the process by giving an example. Let us try to determine the GCF of 16 & 24
Step 1:  Let us write the numbers side by side like this 16, 24
Now we try to find any small integer that will divide both these numbers. We can easily see that both 16 & 24 are even numbers. So, we choose 2 as our divisor.
Step 2:  Now again we try find a common divisor for 8 & 12. So
This process continues till the numbers in the bottom are relatively prime (i.e they have no common divisor). So, we have
Step 3:  To get the GCF we multiply all the common divisors. Note that the common divisors are written on the left.
:. GCF of 16 & 24 = 2x2x2 = 8

 :. GCF of 16 & 24 = 2x2x2 = 8
Another way:
:. GCF of 16 & 24 = 8
Example: Find GCF of 15, 45, 75 & 90.
:. GCF of 15, 45, 75 & 90 = 3x5 = 15

If you find this difficult you can use the prime factorization method. For example, let us try to determined the GCF of 15, 45, 75 & 90.
We will find write down the prime factorization of all the numbers.
15 = 3x5
45 = 3x3x5
75 = 3x5x5
90 = 3x3x2x5
Observe that all the prime factorizations contain at least one '3' & one '5'. So, the GCF = 3x5 = 15

For to see our second lecture Please click on the link below:

Factors, Multiples, GCF & LCM: Multiples, LCM

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