Question

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

Answer 1

We have

Any positive integer is of the form $4q+1or4q+3$

As per Euclid’s Division lemma.

If $a$ and $b$ are two positive integers, then,

$a=bq+r$

Where $0\le r<b$.

Let positive integers be $a$.and $b=4$

Hence,$a=bq+r$

Where, $(0\le r<4)$

R is an integer greater than or equal to $0$ and less than $4$

Hence, $r$ can be either $0,1,2\text{and}\text{3}$

Now, If $r=1$

Then, our be equation is becomes

$a=bq+r$

$a=4q+1$

This will always be odd integer.

Now, If $r=3$

Then, our be equation is becomes

$a=bq+r$

$a=4q+3$

This will always be odd integer.

Hence proved.