Question

Product of two consecutive integer is divisible by 2 is the statement true or false justify

Answer 1

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 (You can try to understand why we take it as this , by giving valies for k as 1,2,3.... that it gives even values)
∴ x²+x= (2k)²+2k
=4k²+2k
=2(2k²+k)

hence divisible by two.

(ii)Let x be odd.
∴ x= 2k+1(You can try to understand why we take it as this , by giving valies for k as 1,2,3.... that it gives odd values)
∴ 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