Question

use Euclid division lemma to show that any positive odd integer is of the form 6q+1 or 6q+3 6q+5 where quality some integer

Answer 1

By euclid's division lemma,
A=bq+r where a and b are co-prime integers and r is greater than or equal to 0 and less than b


Now

Let b=6 then
R =1/2/3/4 or 5

Case 1
A=6q+0
=6q
=2(3q)
Therefore 6q is divisible by 2 amd is even


Case 2
A=6q+1
Therefore 6q+1 is not divisible by 2 and is odd

Similarly
Case3
6q+2=2(3q+1)
Case5
6q+4=2(3q+2)

Case 4
6q+3
And case 6
6q+5



Thus any odd positive integer is of the form 6q+1 or 6q+3 or 6q+5





Hope it helps