Given the statement: For every integer $n,$ if $n$ is odd then $\frac{n-1}{2}$ is odd.

i$.$ Write a negation for the given statement.

ii$.$ Use a counterexample to disprove the given statement. Explain how the counterexample actually shows

that the given statement is false.

Disprove the following statement by giving a counterexample.

For every integer $p,$ if $p$ is prime then $p^2-1$ is even.

Use the method of exhaustion to prove that every positive even integer less than $26$ can be expressed as a

sum of three or fewer perfect squares. $($For instance, $10=1^2+3^2$ and $16=4^2.)$