Math Principles

Friday, October 12, 2012

Greatest Common Factor, GCF

Category: Arithmetic

"Published in Newark, California, USA"

Given the following numbers: 252, 240, 288, and 204. What is their Greatest Common Factor (GCF)?

Solution:

There are two ways in getting their GCF. I will show the both methods and let's see which method is your preference.

Method 1: You can use the intersection method. You have to get their factors of each given numbers.

A = 252 = (1 x 252), (2 x 126), (3 x 84), (4 x 63), (6 x 42), (7 x 36), (9 x 28), (12 x 21), (14 x 18)
A = (1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252)

B = 240 = (1 x 240), (2 x 120), (3 x 80), (4 x 60), (5 x 48), (6 x 40), (8 x 30), (10 x 24), (12 x 20), (16 x 15)
B = (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240)

C = 288 = (1 x 288), (2 x 144), (3 x 96), (4 x 72), (6 x 48), (8 x 36), (9 x 32), (12 x 24), (16 x 18)
C = (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288)

D = 204 = (1 x 204), (2 x 102), (3 x 68), (4 x 51), (6 x 34), (12 x 17)
D = (1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204)

Rewrite the factors of each given numbers,

A = (1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252)
B = (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240)
C = (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288)
D = (1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204)

If A ∩ B ∩ C ∩ D = (1, 2, 3, 4, 6, 12)

Therefore, their GCF is 12.

Method 2: You can use the continuous division method. You have to think their factors as much as you can while dividing the numbers until they can't divide anymore.

2   │ 252 240 288 204
2   │  126 120 144 102
3   │ 63 60 72 51
21 20 24 17

At this stage, 17 is already a prime number. Therefore, the GCF is 2 x 2 x 3 = 12

Note: Please remember the procedure in getting the GCF of the numbers very well because you will use this method later in simplifying a fraction into a lowest term. Also, you must memorize or remember the prime and composite numbers all the time.

Thursday, October 11, 2012

Prime - Composite Numbers

Category: Arithmetic

"Published in Newark, California, USA"

What is the difference between a prime number and a composite number? Well, a prime number is a number that has only 2 factors, the number itself and one. The example of a prime number is 13. On the other hand, a composite number is a number that has 3 or more factors. The example of a composite number is 12. The factors of 12 are 1, 2, 3, 4, 6, and 12. How do you know that a number is a prime number or a composite number? Well, we have a technique or method to determine if a number is a prime number or a composite number which is Sieve of Eratosthenes. This method was imposed by Eratosthenes, a Greek Mathematician to separate the prime numbers and composite numbers using a simple algorithm.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Let's write the numbers from 1 to 100 as shown the table above. Leave 1 as a separate number because 1 is a universal factor. Next, list down the multiples of 2 like 4, 6, 8, 10, 12, 14, and so on. Using a marker, mark all the numbers that are multiples of 2 as shown below

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Next, list down the multiples of 3 like 6, 9, 12, 15, 18, 21, 24, and so on. Using a marker, mark all the numbers that are multiples of 3. Follow the same procedure for multiples of 5, multiples of 7, multiples of 11, multiples of 13, multiples of 17, multiples of 19, and so on. The resulting table will be like this:

  1 2 3   4     5   6     7   8     9   10

11   12   13 14   15   16     17   18   19   20

21   22   23 24   25   26   27 28   29   30

31   32   33 34   35   36   37 38   39   40

41   42   43 44     45   46   47 48   49   50

51   52   53 54   55   56   57 58   59   60

61   62   63 64   65   66   67 68   69   70

71 72   73   74   75   76   77 78   79   80

81 82   83 84     85   86   87 88   89   90

91 92   93    94   95   96   97   98   99   100

The numbers that are not shaded are the prime numbers and the shaded numbers are the composite numbers. You need to memorize or remember the prime numbers and composite numbers very well, if possible because later on, you will use these numbers when you will take higher Math subjects. Also, these numbers will be used later for simplifying a fraction into a lowest term by dividing its common factor.