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Question

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


Solution

 Let $$a$$ be any odd positive integer and $$b=4$$, By division Lemma there exists integer $$q$$ and $$r$$ such that 
$$a=4q+r$$, where $$0\le r < 4$$

$$\Rightarrow \quad a=4q$$ or,$$a=4q+1$$ or $$a=4q+2$$ or$$a=4q+3$$       $$[\because 0 \le r < 4 \Rightarrow r=0,1,2,3]$$

$$\Rightarrow \quad a=4q+1$$ or, $$a=4q+3$$        [$$\because\ a$$ is an odd integer $$\therefore a \neq 4q, a \neq 4q+2]$$

Hence, any odd integer is of the form $$4q+1$$ or, $$4q+3$$.

Mathematics

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