Question

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

Answer 1

Explanation:

I am taking q as some integer.

Let a be the positive integer.

And, b = 4 .

Then by Euclid's division lemma,

We can write
`a = 4q + r` ,for some integer
`q\ \text {and}\ 0 \leq r < 4 .`

°•° Then, possible values of r is 0, 1, 2, 3 .

`\texttt {Taking} \ r = 0 .\\\\a = 4q .`

`\texttt {Taking} \ r = 1 .\\\\a = 4q + 1 .\\\\\texttt {Taking} \ r = 2\\\\a = 4q + 2 .\\\\\texttt {Taking} \ r = 3 .\\\\a = 4q + 3 .`

But a is an odd positive integer, so a can't be 4q , or 4q + 2

[ As these are even ] .

•°• a can be of the form 4q + 1 or 4q + 3 for some integer q .