Question

use euclid division lemma to show that any positive odd integer is of the form 6q+1 or6q+3 or 6q+5, where q is some of the integers

Answer 1

Let us assume that 'a' is an integers. ( 'a' can be either odd or even number ) 

Euclid's division lemma - a = bq+r ( 0 ≤ r < b ) 

Here in this case b=6 
⇒ a = 6q+r ------ equation one 
If b = 6 then - 
(r=0) or (r=1) or (r=2) or (r=3) or (r=4) or (r=5)
 
Let r=0
⇒ a = 6q+r
⇒ a = 6q+0
⇒ a = 6q  
∴ a is even number 

let r=1 
⇒a = 6q+1 
∴ a is odd number 

let r=2 
⇒a = 6q+2 
∴ a is even number 

let r=3 
⇒ a = 6q+3 
∴ a is odd number 

let r=4 
⇒ a = 6q+4 
∴ a is even number 

let r=5 
⇒ a = 6q+5 
∴ a is odd number 

∴ a is an odd positive integer of the form 6q+1 (or) 6q+3 (or) 6q+5 
Hence proved. 

Answer 2

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