Question

Prove that the product of the two consecutive integers is divisible by 2

Answer 1

$\Huge{\red{\underline{\textsf{Solution}}}}$

Let the first integer be x

then the second integer shall be x+1

then their product be x(x+1) = x²+x

(i) If x is even

then x = 2k

∴ x²+x= (2k)²+2k

=4k²+2k

=2(2k²+k)

hence divisible by two.

(ii)Let x be odd.

∴ x= 2k+1

∴ x²+x = (2k+1)²+2k+1

=(2k)²+8k+1+2k+1

=4k²+10k+2

=2(2k²+5k+1)

hence divisible by two.

since bothe of our conditions satisfy the statement, we can say that the product of two consecutive integers is divisible by 2.