Question

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

Answer 1

Euclid's lemma a=bq+r
Let b=6
R can take the values 0, 1, 2, 3, 4, 5
Let r be 1
a=6q+0
a=2(3q)
As a is 'a' multiple of 2 it is an even number.

r=1
a=6q+1
a=2(3q)+1
As 'a' is in the form of 2m+1 it is odd

r=2
a=6q+2
a=2(3q+1)
It is in a multiple of 2 so it is an even number

r=3
a=6q+3
a=2(3q)+3
It is an odd number

r=4
a=6q+4=2(3q+2)
It is an even number

r=5
a=6q+5
It is an odd number
So it is proved that any positive odd integer is of the form 6q+1, 6q+3, 6q+5