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

Given:

The given positive integer is q.

To do:

We have to show that any positive odd integer is of the form 4q$+$1 or 4q$+$3 for some integer 'q'.

Solution:  

By Euclid's division algorithm,

If a and b are two positive integers, then,

$a = b q +r$, where $0 \leq r < b $

Let a be the positive integer, and b$=4$, 

$a = 4 q + r$, where $0 \leq r < 4 $

$r = 0 , 1 , 2 , 3$

Here, 1 , 3 are positive odd integers.

So, possible values of r is 1 ,3.

When $r = 1$,

$a = 4 q + 1$

It is positive odd integer.

When $r = 3$,

$a = 4 q + 3$

It is positive odd integer.

Therefore, any positive odd integer is of the form 4q$+$1 or 4q$+$3 for some integer 'q'.

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Updated on: 10-Oct-2022

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