Question

prove that the product of two consecutive positive integers is divisible by 2

Answer 1

let the two consecutive integers be x and x+1 so their product is x^2+x
if x is positive that 2k then,
(2k)^2+2k is divisible and
if x is negative that is 2k+1...
but substitution in x^2+2 we get that it is also divisible by 2
hence proved

Answer 2

Answer:

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Let n and n + 1 are two consecutive positive integer

We know that n is of the form n = 2q and n + 1 = 2q + 1

n (n + 1) = 2q (2q + 1) = 2 (2q2 + q)

Which is divisible by 2

If n = 2q + 1, then

n (n + 1) = (2q + 1) (2q + 2)

= (2q + 1) x 2(q + 1)

= 2(2q + 1)(q + 1)

Which is also divisible by 2

Hence the product of two consecutive positive integers is divisible by 2