Number System 5

Test of a Prime Number

A prime number is only divisible by 1 and by the number it self. The first prime number is 2. Every prime number other then 2 is odd, but every odd number is not necessarily a prime number. To test whether any given number is a prime number or not, following steps are to be considered:

Step 1:- Find an integer (x) which approximate square root is greater then of the given number.

Step 2:- Test the divisibility of the given number by every prime number less than x.

Step 3:- If the given number is divisible by any of them in Step 2, then the given number is not a prime number.

If the given number is not divisible by any of them in Step 2, then the given number is a prime number.

Ex.:- Consider a number 203. Test if it is a prime or not.

Step 1:- The approximate square root of 203 is 14 plus, take x = 15.

Step 2:- Check the divisibility of 203 by the prime number less than 15 i.e., by 2, 3, 5, 7, 11, 13.

Step 3:- 203 is divisible by 7. Thus, it is not a prime number

SUM rules on Natural Numbers

Rule 1

Sum of the first n natural numbers (Starting from 1) = ^{n(n + 1)}/₂.

Ex.:- Sum of 1 to 74 = ^{74 x 75}/₂ = 2775

Rule 2

Sum of first n odd numbers (Starting from 1) = n²

Ex.:- Sum of first 7 odd numbers ( 1 + 3 + 5 + 7 + 9 + 11 + 13) = 7² = 49

Rule 3

Sum of first n even numbers (Starting from 1) = n(n + 1)

Ex.:- Sum of first 9 even numbers (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18) = 9(9 + 1) = 90.

Rule 4

Sum of squares of first natural numbers (Starting from 1) = ^{n(n + 1)(2n + 1)}/₆

Ex.:- Sum of squares of first 8 natural numbers

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² = ^{8(8 + 1)(2 x 8 + 1)}/⁶

= ^{8 x 9 x 17}/₆ = 204

Rule 5

Sum of cubes of first n natural numbers = [ ^{n(n + 1)}/₂ ]²

Ex.:- Sum of cubes of first 6 natural numbers = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = [ ^{6(6 + 1)}/₂ ]² = 441

Note:- For applying Rule 2 and Rule 3, it is required to find how many odd numbers or even numbers are there in the given series.

In the first ‘n’ natural numbers,

If n is even, then there are ⁿ/₂ odd numbers and ⁿ/₂ even number

If n is odd, then there are ^{n +} ½ odd numbers and ^{n –} ½ even numbers

Ex.:- From 1 to 30, as 30 is even, there are 15 odd numbers and 15 even numbers.

From 1 to 29, as 29 is odd, there are ^{29 +} ½ = 15 odd numbers and ^{29 –} ½ even numbers.