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

By taking,’ a’ as  any positive integer and b = 6.
Applying Euclid’s  algorithm
a = 6q+r                
Here,
So, total possible forms are
6q+0 , (6 is divisible by 2 , its an even number)
6q+1, ( 6 is divisible by 2 but 1 is not divisible by 2 , its an odd number)
6q+2, (6 and 2 both are divisible by 2 , its an even number)
6q+3, (6 is divisible by 2 but 3 is not divisible by 2, its an odd number)
6q+4, ( 6 and 4 both are divisible by 2, its an even number)
6q+5 , (6 is divisible by 2 but 5 is not divisible by 2 , its an odd number)
Therefore,
So, odd numbers will be  in the form 6q+1, or 6q+3, or 6q+5