Question

Show that any positive odd integers is of the form 4q + 1 or 4q+3 . Where q is some integers

Answer 1

Answer :

Let "a" be the positive odd integer which when divided by 4 gives q as quotient and "r" as remainder.

According to Euclid's division lemma :

a = bq + r, where 0 ≤ r < b

Here b = 4 and 0 ≤ r < 4

a = 4q + r

=> a = 4q + 0 (when r = 0)

Even

=> a = 4q + 1 (when r = 1)

Odd

=> a = 4q + 2 (when r = 2)

Even

=> a = 4q + 3 (when r = 3)

Odd

So, a = 4q+1 or, a = 4q+3 are odd

Hence, any positive odd integer is of the form 4q+1, 4q+3

Answer 2

$\huge\mathfrak\blue{Answer:}$

To Show:

Any positive odd integers is of the form 4q + 1 or 4q+3 . Where q is some integers

Solution:

Let a be any positive integer.

By Euclid's Division Lemma,

a = bq + r

So, the possible remainders are 0, 1, 2 and 3.

=> a can be of the form 4q, 4q+1, 4q+2 and 4q+3.

where q is the quotient .

a is odd and hence cannot be of the form 4q or 4q + 2 as they are even.

So, a will be in the form of 4q + 1 or

4q + 3.

Hence proved.