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**The Number Tree****10. Odd and Even Numbers**

- This is a property of Integers. But since our questions are limited to the Whole numbers, we will keep our studies limited to them.
- By Definition,
**Even****Numbers**are those which are divisible by 2. They are the numbers with their unit digit as either 2, 4, 6, 8 or 0. The general expression of Even numbers is 2n where n is a whole number. - By Definition,
**Odd****Numbers**are those which are not divisible by 2. They are the numbers with their unit digit as either 1, 3, 5, 7 or 9. The general expression of Odd numbers is 2n+1 where n is a whole number.

__Certain Operations on Whole Numbers__:**(i)** Odd ± Odd = Even**(ii)** Even ± Even = Even**(iii)** Odd ± Even = Odd**(iv)** Even ± Even ± Even ........... n terms = Even**(v)** Odd ± Odd ± Odd .............. n terms = Even if n is Even, Odd if n is Odd**(vi)** Odd × Odd = Odd**(vii) **Even × Odd = Even**(viii) **Even × Even = Even**(ix) **Even × Even × Even ....... n terms = Even**(x) **Odd × Odd × Odd ....... n terms = Odd**(xi)** (Even)^{n} = Even**(xii)** (odd)^{n} = Odd

Try yourself:If a is even what would 5a + 1 be

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Try yourself:If a is even and b is odd what would a^{3} + 3b + 6 be

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Try yourself:If 5a - 2 is even what would a be

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**11. Prime and Composite Numbers**

**(a) Prime Numbers**: Being Prime is a property of Natural Numbers. All those numbers which are divisible by exactly 2 numbers, i.e. itself and unity "1".**Example:** 2, 3, 5, 7, 11 etc.

**(b) Composite **are the numbers which are divisible by more than 2 numbers, i.e. itself, unity "1" and at least one other.**Example:** 4, 6, 8, 9, 10 etc.

1 is neither Prime nor Composite.

Try yourself:How many Prime numbers are there less than 100

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Try yourself:Which of the following cannot be the sum of two prime numbers.

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Try yourself:State Whether True or False

All Prime numbers are Odd

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Try yourself:State Whether True or False

Sum of Any 2 prime numbers is always Even

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Try yourself:State Whether True or False

Product of Any two prime numbers is always Odd

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Try yourself:State Whether True or False

There are no 4 digit Prime numbers.

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**➢ Property of Prime Numbers**

- All Prime numbers greater than 3 are of the form 6n ± 1 but not all numbers of form 6n ± 1 are Prime. This property means when prime numbers greater than 3 are divided by 6 they leave a remainder of 1 or 5.

Try yourself:Which of the following is a prime numbers.

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**12. Co - Prime Numbers**

- This is a property of a pair or more than two natural numbers.
- Two numbers are said to be co-prime if the only number which can completely divide both the numbers is 1. No number other than 1 can divide both the given numbers of the pair.
- These numbers in themselves can be prime or composite. 1 is co-prime with all the natural numbers other than 1.
**Example:**(3, 7); (4, 15); (3, 10) etc. - A set of numbers is said to be co-prime if their HCF is 1. We will be learning about HCF in the following chapters.
**Example:**(3, 7, 11); (4, 15, 10); (3, 10, 8) etc.

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