Question

Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where Q is some integer

Answer 1

Solution :

Let ' a ' be any positive integer.

Here, b = 4.

By using Euclid's division lemma,

• a = bq + r,  where 0 ≤ r < d.

So possible vales of r = 0, 1, 2 , 3 and 4.

Now,

Case : 1

a = 4q + 0 _______ [ Even ]

Case : 2

a = 4q + 1 ______ [odd]

Case : 3

a = 4q + 2______[Even]

Case : 4

a = 4q + 3 _____[Odd]

Case : 5

a = 4q + 4 ______[Even]

From this, it is clear that any positive odd integer is of the form 4q + 1 or 4q + 3.

Hence proved!

$\rule{200}2$