Which one of the altematives is a proof by contrapositive of the statement If x3-x+4 is not divisible by 4 then x even. 1. Required to prove. If x3-X + 4 is not divisible by 4 then x even. Proof Suppose x is odd. Let x " 2k + 1,then we have to prove that x, x-x+4 (2k 1)- (2k +1)+4 -(2k+1X4k+ 4k +1)-2k-1+4 - 8k+ 8k+2k+4k+ 4k +1 -2k-1+4 -x+ 4 is divisible by 4. 8k3 + 12k2 + 4k + 4 2+3k+ k+ 1), which is divisible by 4. (4 multiplied by any integer is divisible by 4) 2. Required to prove: Ifx3-x+ 4 is not divisible by 4 then x even. Proof Assume that x3-x + 4 is not divisible by 4. Then x can be even or odd we assume that x i odd Let x 2k+ 1, then x3-x+4-(2k+1P-(2k+ 1)+4 - (2k + 1 4k2+4k 1)-2k-1+4 - 8k+ 8k2 + 2k +4k2 4k +1-2k-1+4 8k3 + 12k2 + 4k + 4 4(2k+3k2 k+ 1), which is divisible by 4. (4 multlplied by any integer is divisible by 4) But this is a contradiction to our original assumption. Therefore x must be even ifx3-x+4 is not divisible by 4. 3. Required to prove: Ifx3-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-x4 (4-(4) 4 64-4+ 4 64 which is divisible by 4. Required to prove: lf x3-x + 4 is not divisble by 4 then x even. Proof Assume that x is even, ie x # 4k, then x - x+4 = (4k)-(4k) + 4 -64k-4k +4 4(16k-k + 1), which is divisible by 4 4