Which one of the alternatives is a proof by contrapositive of the statement If x$3$ x $+4$ is not divisible by $4,$ then x even.

a$.$

Required to prove: If x$3$ x $+4$ is not divisible by $4$ then x even.

Proof: Suppose x is odd. Let x $=2$k $+1,$ then we have to prove that x$3$ x $+4$ is divisible by $4.$

x$3$ x $+4=(2$k $+1)3(2$k $+1)+4$

$=(2$k $+1)(4$k$2+4$k $+1)2$k $1+4$

$=8$k$3+8$k$2+2$k $+4$k$2+4$k $+12$k $1+4$

$=8$k$3+12$k$2+4$k $+4=4(2$k$3+3$k$2+$ k $+1),$ which is divisible by $4.(4$ multiplied by any integer is divisible by $4)$

b$.$

Required to prove: If x$3$ x $+4$ is not divisible by $4,$ then x even.

Proof: Assume that x$3$ x $+4$ is not divisible by $4.$

Then x can be even or odd. We assume that x is odd.

Let x $=2$k $+1,$ then x$3$ x $+4$

$=(2$k$+1)3(2$k $+1)+4$

$=(2$k $+1)(4$k$2+4$k $+1)2$k $1+4$

$=8$k$3+8$k$2+2$k $+4$k$2+4$k $+12$k $1+4$

$=8$k$3+12$k$2+4$k $+4$

$=4(2$k$3+3$k$2+$ k $+1),$ which is divisible by $4.(4$ multiplied by any integer is divisible by $4)$

But this is a contradiction to our original assumption. Therefore x must be even if x$3$ x $+4$ is not divisible by $4.$

c$.$

Required to prove: If x$3$ x $+4$ is not divisible by $4,$ then x even.

Proof: Let x $=4$ be an even element of Z$.$ We can replace x with $4$ in the expression x$3$ x $+4.$

x$3$ x $+4=644+4=64$ which is divisible by $4.$

d$.$

Required to prove: If x$3$ x $+4$ is not divisible by $4,$ then x even.

Proof: Assume that x is even, i$.$e$.$ x $=4$k$,$ then

x$3$ x $+4=(4$k$)3(4$k$)+4=64$k$34$k $+4=4(16$k$3$ k $+1),$ which is divisible by $4.$