Question

show that every positive even integer is of the form2q and that every poaative odd integer is of the form 2q+1 where q is some integer

Answer 1

let a be a positive integer
when it is divided by 2
then by euclid's division Lemma
we have a=2q+r
where 0≤r<2
so r=0,1
therefore a either equal to 2q or 2q+1
so 2q is divided by 2 and is even where as in case of odd it is opposite.

Answer 2

Heya user !!!

Here's the answer you are looking for

Let us assume the number is say, x

By Euclid's lemma, x can be represented as

x = 2a + r ( where a is any integer and 0≤ r < 2)

So, the value of r can be either 0 or 1.

CASE 1
When r = 0

x = 2a

Since x is a multiple of 2, it is even. And hence every even positive number can be represented as 2q ( where q is any integer)


CASE 2
when r = 1

x = 2x + 1

x is not divisible by 2, so it is odd. And hence every odd number can be represented as 2q + 1 (where q is any integer)



★★ HOPE THAT HELPS ☺️ ★★