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
Question for Important Concepts: Number System - 2
Try yourself:If a is even what would 5a + 1 be
Explanation
Since a is even and 5 is odd
so 5a will be Even × Odd = Even
1 is odd
So, 5a + 1 will be Even + Odd = Odd
Question for Important Concepts: Number System - 2
Try yourself:If a is even and b is odd what would a^{3} + 3b + 6 be
Explanation
Since a is even and a^{3} is Even^{3} = Even
b is odd so 3b is Odd × Odd = Odd
6 is Even
So, a^{3} + 3b + 6 becomes Even + Odd + Even = Odd.
Question for Important Concepts: Number System - 2
Try yourself:If 5a - 2 is even what would a be
Explanation
There are two cases i.e. a can be odd or even
When a is odd, 5a will be odd × odd = odd
So 5a - 2 is Odd - Even = Odd
When a is even, 5a will be odd × even = even
So 5a - 2 is Even - Even = Even
So a has to be even for 5a - 2 to be even.
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.
Question for Important Concepts: Number System - 2
Try yourself:How many Prime numbers are there less than 100
Explanation
Following is the list of all the prime numbers less than 100
Question for Important Concepts: Number System - 2
Try yourself:Which of the following cannot be the sum of two prime numbers.
Explanation
63 = 61 + 2
100 = 97 + 3
36 = 19 + 17
Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False All Prime numbers are Odd
Explanation
2 is an Even number as well as Prime
Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False Sum of Any 2 prime numbers is always Even
Explanation
Let the two prime numbers be 2 and 3. Their sum is 5 which is odd.
Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False Product of Any two prime numbers is always Odd
Explanation
Let the two prime numbers be 2 and 3. Their product is 6 which is even.
Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False There are no 4 digit Prime numbers.
Explanation
1009 is a prime number, rather there are 1061, 4 digit prime numbers that exist.
➢ 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.
Question for Important Concepts: Number System - 2
Try yourself:Which of the following is a prime numbers.
Explanation
When 9929(A) is divided by 6 remainder is 5
When 9939(B) is divided by 6 remainder is 3
When 9998(C) is divided by 6 remainder is 2
When 9995(D) is divided by 6 remainder is 5 but this number is divisible by 5
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.