Question

5. 20 pts Prove that if n is an odd positive integer, then n 1 (mod 8)

this needs to be proven using a direct proof, proof by contraposition, or a proof by contradiction. YOU MUST SHOW ALL STEPS LABEL THEM AND SPACE THEM WELL PLEASE. ALSO INCLUDE ALL DEFINITIONS USED . FORMAL PROOFS ONLY PLEASE

Answer 1

Let i be an integer number

Even Integer number =

Odd Integer Number =

where $i=1,2,3,\dots$

Odd Integer Number $n=2i+1$

$n^2=(2i+1)^2 \newline n^2 = (2i+1)^2 = 4i^2 +4i +1 \newline (n^2 - 1) = 4i(i +1)$

If i is an even number then i(i+1) is also even value

If i is an odd number then i(i+1) is also even value

So here $l\epsilon Z$ such that

$n^2-1 = 4 i (i+1) = 4\times 2l =8l$

$n^2=8l+1$

Taking Mod 8 on Both Sides

$n^2(mod\hspace{1em} 8)=(8l+1)(mod \hspace{1em} 8) \newline =(8l)(mod \hspace{1em} 8) + 1(mod \hspace{1em} 8) \newline =0 (mod \hspace{1em} 8) + 1(mod \hspace{1em} 8) \newline \equiv 1(mod \hspace{1em} 8)$