Prime Factors

Prime Factors

We are just finding to be listing out the factors of number. Don’t panic! It is very easy way to find factors of number and prime factors.

Remember when you do a multiplication exercise … …

2 × 3 = 6

Here, 2 and 3 are called factors. 6 is the product.

So, to find all the factors of 6, we just want to list out all possible ways that we can get 6 using whole numbers and multiplying.

1 ×  6 = 6

2 ×  3 = 6

Hence, the factors of 6 are 1, 2, 3, 6.  I’m just listing them in orderly.

Let’s try another example 18.

1 ×  18 = 18

2 ×  9 = 18

3 ×  6= 18

Notice that I’m just working through all number that result 18.

When I hit 3 × 6= 18, It would be too silly go on because the next pair will be 4 ×  3 = 18 and that’s just repeat.

So, the factors of 18 are

1, 2, 3, 6, 9, 18.

Find the factor of 72.

1 ×  72 = 72

2 ×  36 = 72

3 ×  24 = 72

4 ×  18 = 72

6 ×  12 = 72

8 ×  9 = 72

The factors of 72 are …

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

OK, one more:

Find the factor of 19.

1 ×  19 = 19

Yep… Looks like that’s it!

The factors of 19 are 1 and 19.

YOUR TASK:

Find the factor of 15.

Find the factor of 20.

Find the factor of 35.

Find the factor of 54.

Find the factor of 63.

Find the factor of 85.

Prime and Composite Numbers

1 ×  19 = 19

1 ×  23 = 23

Did you notice that about the above numbers 19 and 23?

There was only one way to get them i.e., they each only had two factors.

When this happens, it is called prime number.

A prime number has exactly two factors 1 and itself.

If a number is not a prime number then it is called a composite number.

Officially, a composite number has more than two positive factors.

So, if you can find anything else that goes into a number other than 1 and itself, then it is called a composite number.

Is 798 a composite number?

Yes! It is even, so 2 divide it.

By the way, number 1 is special. It only has one factor 1 i.e., just itself.

Hence, 1 is neither prime nor composite.

YOUR TASK:

Find the ten prime and ten composite numbers and write down your notebook.

Eratosthenes method

Now, we discuss about a method that came up with a mathematical way to find prime numbers.

This method name is Eratosthenes method.

Don’t worry about! It is very easy method.

Here’s this method:

You start by making a grid of the numbers 1 to 100:

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

Now we list all prime numbers 1 to 10 i.e., 2, 3, 5, 7 and

cross out all number which are divide by 2, 3, 5 ,7.

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

Therefore, the prime numbers that exist between 1 and 100 are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Prime Factorization

We know to learn about Factorization

24 = 3 ×  8

But, can we write 18 as the product of only prime numbers?

Well, 3 is a prime number, but 8 isn’t?

Can we rearrange 8 into product of prime numbers?

8= 2 ×  2 ×  2

So, 24 = 2 ×  2 ×  2 ×  3

i.e. if we write a number as the product of primes then it’s called a prime factorization.

Now it is very easy for you.

Let’s try another.

Find the prime factorization of 72:

Use time tables

72 = 3 ×  24

Here, 3 is a prime number, but 24 isn’t?

Can we rearrange 24 into product of prime numbers?

24 = 2 ×  2 ×  2 ×  3 (use previous example)

Therefore, 72 = 2 ×  2 ×  2 ×  3 ×  3

Here it is with exponents form:

72 = 23 × 3 2

Some number are very

15 = 3 ×  5

Now you turn.

Find the prime factorization of 48?

Factor tree:

Prime Factorization