CBSE Notes, Lectures

CBSE - Mathematics - Real Numbers

Real Numbers

NCERT Exercise Exercise 1.1

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Let take a as any positive integer and b = 6.

Then using Euclid’s algorithm we get a = 6q + r here r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2, 3, 4, 5 because 0 ≤ r < b and the value of b is 6 
So total possible forms will 6q + 0 , 6q + 1 , 6q + 2,6q + 3, 6q + 4, 6q + 5

6q + 0
6 is divisible by 2 so it is a even number 

6q + 1 
6 is divisible by 2 but 1 is not divisible by 2 so it is a odd number 

6q + 2 
6 is divisible by 2 and 2 is also divisible by 2 so it is a even number 

6q  +3 
6 is divisible by 2 but 3 is not divisible by 2 so it is a odd number 

6q + 4 
6 is divisible by 2 and 4 is also divisible by 2 it is a even number
 
6q + 5 
6 is divisible by 2 but 5 is not divisible by 2 so it is a odd number

So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5
.