Question

use Euclid's devision lemma to show that any poditive odd integer is of form 6q+1 or 6q+3 or 6q+5, where qis some integer

Answer 1

let q is a integer
b=6 r=0,1,2,3,4,5
then apply E. D. l
a=bq+r
1.a=6q+0=6q
2.a=6q+1
3.a=6q+3
4.a=6q+4
5.a=6q+5
here take only odd integer.
these are 6q+1,6q+3,6q+5. are odd integer


I hope it helps your

Answer 2

Let b= 6 then possible remainder (r) =0,1,2,3,4,5. now positive odd integers are 1,3,5. a= 6q+1 ,a=6q+3,a= 6q+5 from a= 6q+1 = 2×3q+1. = 2k+1(is in the form of 2 q +1 ). a= 6q+3. = 2×3q+2+1. 2(3q+1)+1. = 2k +1 ( is in the form of 2q+1) a= 6q+5. =6q+4+1 =2×3q+2×2+1. =2(3q+2)+1. = 2k+1 ( is in the form of 2q+1). from above all are in the form of 6q+1or 6q+3 or 6q+5