CHAPTER 3: Playing with Numbers

Prime Numbers and Factors

Prime Numbers

Prime numbers are numbers which cannot be divided by any number except itself and 1. So in

1 to 10 we have prime numbers 2, 3, 5, and 7.  These numbers are not divisible by any number between 1 to 10.

1) Now for numbers between 1 - 10 x 10 (100) : we determine prime numbers if they are divisible by 2,3,5 and 7 for number from 1 to 100

2) Now for numbers between 11 - 20 x 20 (400): we determine prime numbers if they are divisible by 2,3,5,7,11, 13, 17, 19 (these are prime numbers between 11 to 20) for numbers between 100 - 400

3) Now for number between 21 - 30 x30 (900) : we determine prime numbers if they are divisible by 2,3,5,7,11,13, 17,19,23 and 29 (these are prime numbers between 20 and 30) for numbers between 400 - 900

So on you can determine prime numbers to million number.

The above paragraph means the following:

In 1 to 10 the prime factors are 2,3,5, and 7. These prime factors can determine prime factors from 1 to 100 ( 10 x 10)

In 1 to 20 the prime factors are 2,3,5, 7 , 11, 13, 17 and 19. These prime factors can determine prime factors from 1 to 400 ( 20 x 20)

In 1 to 30 the prime factors are 2,3,5, 7 , 11, 13, 17 ,19 , 23, 29. These prime factors can determine prime factors from 1 to 900 ( 30 x 30)

If the prime numbers cannot divide a given number then that number is a prime number.

Factors

Example 1:

If you are given number say 60 and told to find factors of 60 then one rule i can tell you is that factors will be upto 60/2 = 30

So from 1 to 30 you have to find numbers which are factors of 30.

Now do prime factorization of 60:

 2 | 60

 2 | 30 

 3 | 15 

5  | 5

    | 1 

Now prime numbers 1, 2, 3, 5 are factors of 60. Inside the division box 15 and 30 are also factors of 60. Now start from top outside the

division box i.e. consider 2,2, 3, 5 from multiplying these numbers with each other we get factors of 60. So

2 x 2 = 4 which is a factor of 60

2 x 2 x 3 = 12 which is a factor of 60

2 x 2 x 3 x 5 = 60 which is the factor of 60

3 x2 = 6 which is a factor of 60

Now you go from bottom up:

5 x3 = 15 which we have already considered it.

5 x3 x 2 = 30 which we have already considered it

5 x 3 x 2 x 2 = 60 which we have already considered it

5 x 2 = 10 which we have not considered

5 x 2 x 2 = 20 which we have not considered

So what are factors of 60, arranging in order we get 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

So essentially what we are doing is multiplying every possible combinations to find the factors.

Example 2:

Find the factor of 126.

Step 1: 126/2 = 63 therefore factors would be from 1 to 63.

Step 2: Do prime factorization of 126:

  2   | 126

  3   | 63 

 3    | 21 

 7    | 7 

      |   1 

1) Now consider numbers outside division box 2, 3, and 7 are factors of 126.

2) Numbers in Division box: 63 and 21 which is a factor of 126

3) Now multiply each and every combination of numbers outside division box we get

a) 7 x 3 = 21 which we have already considered

b)  7 x 3 x 3 =  63 which we have already considered

c) 7 x 3 x 3 x 2 = 126 which we have already considered

d) 3 x 3 = 9 which we have not considered

e) 3 x 3 x 2 = 18 which we have not considered

f) 2 x 3 = 6 which we have not considered

g) 7 x 2 = 14 which we have not considered

h) 7 x 2 x 3 = 42 which we have not considered

Therefore factors of 126 is 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126