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## Here's the Solution to this Question

$a)$

$\text{An error has been made.}$

$\text{A general statement is derived from a particular statement. }$

$if \ n^2\vdots4\ then\ n\vdots4$

$\text{Counter-example:}$

$6^2=36\vdots4 - true; 6\vdots4-false$

$\text{Statement If n is an integer and } n^2 \text{ is divisible by 4, then n is divisible by 4 is false}$

$b)$

$\text{ Statement: Let p and q be integers and r = pq + p + q, then r is even if and only}$

$\text{if p and q are both even is true.}$

$\text{But for completeness of the proof, it is necessary to prove that}$

$\text{If one or both of the numbers p, q are odd then r is odd.}$

$\text{Let one of the numbers p, q be odd then:}$

$pq -\text {is even number}$

$p+q \text { is odd number}$

$pq +p+q \text { is odd number}$

$r \text { is odd number}$

$\text{Let both numbers p and q be odd then:}$

$pq -\text {is odd number}$

$p+q \text { is even number}$

$pq +p+q \text { is odd number}$

$r \text { is odd number}$