Question

3)Show that every positive odd integer is of the form(4q+1) or (4q+3) for some integer q.

Answer 1

According to Euclid's division lemma,

a = bq + r and 0 ≤ r < b

let a= Some integer

b= 4

r= 0,1,2,3

a= 4q, 4q+1, 4q+2, 4q+3

Therefore, a is a positive integer if

a= 4q+1, 4q+3

Answer 2

It says that positive odd integer

by applying euclids division algorithm

$0 \leqslant r \leqslant b$

where r is remainder and b is divisor .

so b=4 ,according the question.

the value of r =1,2,3.

as a=bq+r

the integers will be 4q+1, 4q+2, 4q+3.

As it says positive odd integer

4q+1 and 4q+3 are odd integers.

as the remainders are odd

4q+1 and 4q+3 are odd positive integers.

HENCE PROVED.