Question

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

Answer 1

Let take a as any positive integer and b=6

Then using Euclid's algorithm we get

$a=6q+r$

here $r$ is remainder and value of $q\ge 0$

and $r=0,1,2,3,4,5$ because $0\le r<b$ and the value $b=6$

So total possible forms will be $6q+0,6q+1,6q+2,6q+3,6q+4,6q+5$

$6q+0$

6 is divisible by 2 so it is even number

$6q+1$

6 is divisible by 2 but 1 is not divisible by 2 so it is odd number

$6q+2$

6 is divisible by 2 and 2 is also divisible by 2 so it is even number

$6q+3$

6 is divisible by 2 but 3 is not divisible by 2 so it is odd number

$6q+4$

6 is divisible by 2 and 4 is also divisible by 2 so it is even number

$6q+5$

6 is divisible by 2 but 5 is not divisible by 2 so it is odd number

So, odd numbers will in form of $6q+1,6q+3,6q+5.$