For the following, find if there are any errors in the methods of proof given below. List out these errors and write how you would prove/disprove the statements given below. (a) Statement: If n is an integer and n^2 is divisible by 4, then n is divisible by 4. Proof: Consider the number 144, which is a perfect square divisible by 4 ( since 4 × 36 = 144). Now, considering that √ 144 = 12 so n=12. Since 12 is also divisible by 4 (4 × 3 = 12), the statement holds true. Hence, Proved! (b) Statement: Let p and q be integers and r = pq + p + q, then r is even if and only if p and q are both even. Proof: Since p and q are even we can write them as p = 2k1 and q = 2k2. This means - r = 2k1 · 2k2 + 2k1 + 2k2, r = 2(2 · k1 · k2 + k1 + k2), r = 2(k3) Meaning r is an even number. Therefore, the statement above is true.

$a)$

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

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

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

$\text{Counter-example:}$

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

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

$b)$

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

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

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

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

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

$pq -\text {is even number}$

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

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

$r \text { is odd number}$

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

$pq -\text {is odd number}$

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

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

$r \text { is odd number}$